{"id":127,"date":"2024-10-16T18:40:20","date_gmt":"2024-10-16T18:40:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/?post_type=chapter&#038;p=127"},"modified":"2024-10-21T13:50:47","modified_gmt":"2024-10-21T13:50:47","slug":"factoring-polynomials-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/factoring-polynomials-fresh-take\/","title":{"raw":"Factoring Polynomials: Fresh Take","rendered":"Factoring Polynomials: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Factor polynomial expressions using the Greatest Common Factor (GCF) and by grouping to simplify expressions.<\/li>\r\n \t<li>Factor trinomials and perfect square trinomials into binomials.<\/li>\r\n \t<li>Break down expressions like differences of squares and cubic equations into their simpler factors.<\/li>\r\n \t<li>Use specific methods to factor expressions that contain fractional or negative exponents.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>What is factoring?<\/h2>\r\nFactoring in mathematics is the process of breaking down an expression into simpler parts, or \"factors,\" that, when multiplied together, produce the original expression. This method is primarily used in algebra to simplify expressions, solve equations, and analyze functions.\r\n\r\nImagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in the figure below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21210700\/CNX_CAT_Figure_01_05_001.jpg\" alt=\"A large rectangle with smaller squares and a rectangle inside. The length of the outer rectangle is 6x and the width is 10x. The side length of the squares is 4 and the height of the width of the inner rectangle is 4.\" width=\"487\" height=\"259\" \/>\r\n\r\nThe area of the entire region can be found using the formula for the area of a rectangle.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill A&amp; =&amp; lw\\hfill \\\\ &amp; =&amp; 10x\\cdot 6x\\hfill \\\\ &amp; =&amp; 60{x}^{2}{\\text{ units}}^{2}\\hfill \\end{array}[\/latex]<\/p>\r\nThe areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area of [latex]A={s}^{2}={4}^{2}=16[\/latex] units<sup>2<\/sup>. The other rectangular region has one side of length [latex]10x - 8[\/latex] and one side of length [latex]4[\/latex], giving an area of [latex]A=lw=4\\left(10x - 8\\right)=40x - 32[\/latex] units<sup>2<\/sup>. So the region that must be subtracted has an area of [latex]2\\left(16\\right)+40x - 32=40x[\/latex] units<sup>2<\/sup>.\r\n\r\nThe area of the region that requires grass seed is found by subtracting [latex]60{x}^{2}-40x[\/latex] units<sup>2<\/sup>. This area can also be expressed in factored form as [latex]20x\\left(3x - 2\\right)[\/latex] units<sup>2<\/sup>. We can confirm that this is an equivalent expression by multiplying.\r\n\r\nMany polynomial expressions can be written in simpler forms by factoring.\r\n<h2>Factoring the Greatest Common Factor (GCF)<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\n&nbsp;\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Factoring:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Breaking down a polynomial into simpler parts<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Products of these parts equal the original polynomial<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Greatest Common Factor (GCF):\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Largest factor common to all terms in a polynomial<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Includes both numerical coefficients and variables<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Steps for Factoring using GCF:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify GCF of numerical coefficients<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify GCF of variables (lowest exponent for each variable)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine numerical and variable GCFs<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor out the combined GCF<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Distributive Property in Reverse:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]ab + ac = a(b + c)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Used to factor out the GCF<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Importance of Factoring:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Simplifies expressions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Aids in solving equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Helps in understanding algebraic structures<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Factor the following:\r\n<p style=\"text-align: center;\">[latex]25b^3 + 10b^2[\/latex]<\/p>\r\n[reveal-answer q=\"910403\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"910403\"]\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">Identify GCF of coefficients: GCF of [latex]25[\/latex] and [latex]10[\/latex] is [latex]5[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify GCF of variables: Lowest power of [latex]b[\/latex] is [latex]b^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combined GCF: [latex]5b^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor out GCF: [latex]\\begin{align} 25b^3 + 10b^2 &amp;= 5b^2(5b) + 5b^2(2) \\ &amp;= 5b^2(5b + 2) \\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video we see two more examples of how to find and factor the GCF from binomials.<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/25_f_mVab_4?si=TQZjEH1W15I8T_hx\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<\/section>\r\n<h2>Factoring a Trinomial with Leading Coefficient of 1<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Standard Form:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The trinomial is in the form [latex]x^2 + bx + c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This factors into [latex](x + p)(x + q)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Coefficient Relationships:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The sum of p and q equals b: [latex]p + q = b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The product of p and q equals c: [latex]pq = c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor Identification Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Find two numbers that multiply to give c<\/li>\r\n \t<li class=\"whitespace-normal break-words\">These same two numbers should add to give b<\/li>\r\n \t<li class=\"whitespace-normal break-words\">These numbers become p and q in the factored form<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Sign Considerations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If c is positive, p and q have the same sign as b<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If c is negative, p and q have opposite signs, with the larger magnitude having the same sign as b<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verification:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The factored form [latex](x + p)(x + q)[\/latex] should expand back to [latex]x^2 + bx + c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]{x}^{2}-7x+6[\/latex].[reveal-answer q=\"823058\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"823058\"][latex]\\left(x - 6\\right)\\left(x - 1\\right)[\/latex][\/hidden-answer]<\/section>\r\n<h2>Factoring by Grouping (Factoring a Trinomial with Leading Coefficient of Not 1)<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Applicable Trinomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used for trinomials in the form [latex]ax^2 + bx + c[\/latex] where [latex]a \\neq 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Coefficient Relationship:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Need to find two numbers m and n such that [latex]m \\times n = a \\times c[\/latex] and [latex]m + n = b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rewriting the Trinomial:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Express the middle term bx as the sum of two terms: [latex]mx + nx[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rewrite the trinomial as a four-term polynomial: [latex]ax^2 + mx + nx + c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Grouping Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Group the first two and last two terms: [latex](ax^2 + mx) + (nx + c)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor out the GCF from each group<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Common Binomial Extraction:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">After grouping and factoring, a common binomial factor should emerge<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor out this common binomial to complete the factorization<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Final Factored Form:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The result will be the product of two binomials: [latex](px + q)(rx + s)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Where [latex]pr = a[\/latex] and [latex]qs = c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verification:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Multiplying out [latex](px + q)(rx + s)[\/latex] should yield the original trinomial [latex]ax^2 + bx + c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]5x^2 + 7x - 6[\/latex][reveal-answer q=\"893931\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"893931\"]Put Answer Here[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see another example of how to factor a trinomial by grouping.[embed]https:\/\/youtu.be\/agDaQ_cZnNc[\/embed]\r\n\r\n<\/section>\r\n<h2>Factoring a Perfect Square Trinomial<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A perfect square trinomial is a trinomial that can be written as the square of a binomial<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Forms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a^2 + 2ab + b^2 = (a + b)^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a^2 - 2ab + b^2 = (a - b)^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identifying Perfect Square Trinomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">First and last terms are perfect squares<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Middle term is twice the product of the square roots of the first and last terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factoring Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the squares of [latex]a[\/latex] and [latex]b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify the middle term is [latex]2ab[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write as [latex](a \\pm b)^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]16x^2 - 24x + 9[\/latex][reveal-answer q=\"423709\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"423709\"]\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">Identify squares: [latex]\\begin{align*} 16x^2 &amp;= (4x)^2 \\ 9 &amp;= 3^2 \\end{align*}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check middle term: [latex]\\begin{align*} -24x &amp;= 2(4x)(3) \\ -24x &amp;= -24x \\quad \\text{(verified)} \\end{align*}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write factored form: [latex]\\begin{align*} 16x^2 - 24x + 9 &amp;= (4x - 3)^2 \\end{align*}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Factoring a Difference of Squares<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A difference of squares is an expression of the form [latex]a^2 - b^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Form:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a^2 - b^2 = (a+b)(a-b)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identifying a Difference of Squares:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Two terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Both terms are perfect squares<\/li>\r\n \t<li class=\"whitespace-normal break-words\">One term is subtracted from the other<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factoring Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the squares of [latex]a[\/latex] and [latex]b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write as \u00a0[latex]\\left(a+b\\right)\\left(a-b\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]49y^2 - 64[\/latex][reveal-answer q=\"106877\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"106877\"]\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">Identify squares: [latex]\\begin{align*} 49y^2 &amp;= (7y)^2 \\ 64 &amp;= 8^2 \\end{align*}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the difference of squares formula: [latex]\\begin{align*} 49y^2 - 64 &amp;= (7y)^2 - 8^2 \\ &amp;= (7y+8)(7y-8) \\end{align*}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see another example of how to factor a difference of squares.<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Li9IBp5HrFA?si=89OEOlaK8aD-GrdB\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<\/section>\r\n<h2>Factoring the Sum and Difference of Cubes<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Sum of Cubes Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a^3 + b^3 = (a + b)(a^2 - ab + b^2)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Difference of Cubes Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]a^3 - b^3 = (a - b)(a^2 + ab + b^2)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">SOAP Mnemonic:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>S<\/strong>ame sign in first factor<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>O<\/strong>pposite sign between terms in second factor<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>A<\/strong>lways positive for last term in second factor<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>P<\/strong>ositive sign between factors<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Recognition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify expressions in the form [latex]a^3 \\pm b^3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Confirm that both terms are perfect cubes<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Factor the sum of cubes [latex]216{a}^{3}+{b}^{3}[\/latex].[reveal-answer q=\"496204\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"496204\"][latex]\\left(6a+b\\right)\\left(36{a}^{2}-6ab+{b}^{2}\\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Factor the difference of cubes: [latex]1,000{x}^{3}-1[\/latex].[reveal-answer q=\"510077\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"510077\"][latex]\\left(10x - 1\\right)\\left(100{x}^{2}+10x+1\\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following two video examples we show more binomials that can be factored as a sum or difference of cubes.<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tFSEpOB262M?si=6SMNIzv3tugfjC9x\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/J_0ctMrl5_0?si=KHKyUENKhs8_o94U\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<\/section>\r\n<h2>Factoring Expressions with Fractional or Negative Exponents<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">General Approach:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the Greatest Common Factor (GCF) with fractional or negative exponents<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor out the GCF, including the variable with the lowest exponent<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Fractional Exponents:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Treat fractional exponents similarly to integer exponents<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor out the variable with the lowest fractional exponent<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Negative Exponents:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Include negative exponents in the factoring process<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor out the term with the most negative (or least positive) exponent<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combining Terms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">After factoring, combine like terms within parentheses if possible<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]2{\\left(5a - 1\\right)}^{\\frac{3}{4}}+7a{\\left(5a - 1\\right)}^{-\\frac{1}{4}}[\/latex].[reveal-answer q=\"989124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"989124\"][latex]{\\left(5a - 1\\right)}^{-\\frac{1}{4}}\\left(17a - 2\\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video provides more examples of how to factor expressions with negative exponents.<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/4w99g0GZOCk?si=aUnt9qbNJbTpT6eq\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video provides more examples of how to factor expressions with fractional exponents.<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/R6BzjR2O4z8?si=PFDc9vazfllfRnnK\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Factor polynomial expressions using the Greatest Common Factor (GCF) and by grouping to simplify expressions.<\/li>\n<li>Factor trinomials and perfect square trinomials into binomials.<\/li>\n<li>Break down expressions like differences of squares and cubic equations into their simpler factors.<\/li>\n<li>Use specific methods to factor expressions that contain fractional or negative exponents.<\/li>\n<\/ul>\n<\/section>\n<h2>What is factoring?<\/h2>\n<p>Factoring in mathematics is the process of breaking down an expression into simpler parts, or &#8220;factors,&#8221; that, when multiplied together, produce the original expression. This method is primarily used in algebra to simplify expressions, solve equations, and analyze functions.<\/p>\n<p>Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in the figure below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21210700\/CNX_CAT_Figure_01_05_001.jpg\" alt=\"A large rectangle with smaller squares and a rectangle inside. The length of the outer rectangle is 6x and the width is 10x. The side length of the squares is 4 and the height of the width of the inner rectangle is 4.\" width=\"487\" height=\"259\" \/><\/p>\n<p>The area of the entire region can be found using the formula for the area of a rectangle.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill A& =& lw\\hfill \\\\ & =& 10x\\cdot 6x\\hfill \\\\ & =& 60{x}^{2}{\\text{ units}}^{2}\\hfill \\end{array}[\/latex]<\/p>\n<p>The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area of [latex]A={s}^{2}={4}^{2}=16[\/latex] units<sup>2<\/sup>. The other rectangular region has one side of length [latex]10x - 8[\/latex] and one side of length [latex]4[\/latex], giving an area of [latex]A=lw=4\\left(10x - 8\\right)=40x - 32[\/latex] units<sup>2<\/sup>. So the region that must be subtracted has an area of [latex]2\\left(16\\right)+40x - 32=40x[\/latex] units<sup>2<\/sup>.<\/p>\n<p>The area of the region that requires grass seed is found by subtracting [latex]60{x}^{2}-40x[\/latex] units<sup>2<\/sup>. This area can also be expressed in factored form as [latex]20x\\left(3x - 2\\right)[\/latex] units<sup>2<\/sup>. We can confirm that this is an equivalent expression by multiplying.<\/p>\n<p>Many polynomial expressions can be written in simpler forms by factoring.<\/p>\n<h2>Factoring the Greatest Common Factor (GCF)<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Factoring:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Breaking down a polynomial into simpler parts<\/li>\n<li class=\"whitespace-normal break-words\">Products of these parts equal the original polynomial<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Greatest Common Factor (GCF):\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Largest factor common to all terms in a polynomial<\/li>\n<li class=\"whitespace-normal break-words\">Includes both numerical coefficients and variables<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Steps for Factoring using GCF:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify GCF of numerical coefficients<\/li>\n<li class=\"whitespace-normal break-words\">Identify GCF of variables (lowest exponent for each variable)<\/li>\n<li class=\"whitespace-normal break-words\">Combine numerical and variable GCFs<\/li>\n<li class=\"whitespace-normal break-words\">Factor out the combined GCF<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Distributive Property in Reverse:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]ab + ac = a(b + c)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Used to factor out the GCF<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Importance of Factoring:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Simplifies expressions<\/li>\n<li class=\"whitespace-normal break-words\">Aids in solving equations<\/li>\n<li class=\"whitespace-normal break-words\">Helps in understanding algebraic structures<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Factor the following:<\/p>\n<p style=\"text-align: center;\">[latex]25b^3 + 10b^2[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q910403\">Show Answer<\/button><\/p>\n<div id=\"q910403\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Identify GCF of coefficients: GCF of [latex]25[\/latex] and [latex]10[\/latex] is [latex]5[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Identify GCF of variables: Lowest power of [latex]b[\/latex] is [latex]b^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Combined GCF: [latex]5b^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Factor out GCF: [latex]\\begin{align} 25b^3 + 10b^2 &= 5b^2(5b) + 5b^2(2) \\ &= 5b^2(5b + 2) \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video we see two more examples of how to find and factor the GCF from binomials.<iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/25_f_mVab_4?si=TQZjEH1W15I8T_hx\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n<h2>Factoring a Trinomial with Leading Coefficient of 1<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Standard Form:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The trinomial is in the form [latex]x^2 + bx + c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">This factors into [latex](x + p)(x + q)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Coefficient Relationships:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The sum of p and q equals b: [latex]p + q = b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">The product of p and q equals c: [latex]pq = c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Factor Identification Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find two numbers that multiply to give c<\/li>\n<li class=\"whitespace-normal break-words\">These same two numbers should add to give b<\/li>\n<li class=\"whitespace-normal break-words\">These numbers become p and q in the factored form<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Sign Considerations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If c is positive, p and q have the same sign as b<\/li>\n<li class=\"whitespace-normal break-words\">If c is negative, p and q have opposite signs, with the larger magnitude having the same sign as b<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verification:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The factored form [latex](x + p)(x + q)[\/latex] should expand back to [latex]x^2 + bx + c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]{x}^{2}-7x+6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q823058\">Show Solution<\/button><\/p>\n<div id=\"q823058\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(x - 6\\right)\\left(x - 1\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Factoring by Grouping (Factoring a Trinomial with Leading Coefficient of Not 1)<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Applicable Trinomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used for trinomials in the form [latex]ax^2 + bx + c[\/latex] where [latex]a \\neq 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Coefficient Relationship:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Need to find two numbers m and n such that [latex]m \\times n = a \\times c[\/latex] and [latex]m + n = b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Rewriting the Trinomial:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Express the middle term bx as the sum of two terms: [latex]mx + nx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Rewrite the trinomial as a four-term polynomial: [latex]ax^2 + mx + nx + c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Grouping Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Group the first two and last two terms: [latex](ax^2 + mx) + (nx + c)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Factor out the GCF from each group<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Common Binomial Extraction:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">After grouping and factoring, a common binomial factor should emerge<\/li>\n<li class=\"whitespace-normal break-words\">Factor out this common binomial to complete the factorization<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Final Factored Form:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The result will be the product of two binomials: [latex](px + q)(rx + s)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Where [latex]pr = a[\/latex] and [latex]qs = c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verification:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Multiplying out [latex](px + q)(rx + s)[\/latex] should yield the original trinomial [latex]ax^2 + bx + c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]5x^2 + 7x - 6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q893931\">Show Answer<\/button><\/p>\n<div id=\"q893931\" class=\"hidden-answer\" style=\"display: none\">Put Answer Here<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see another example of how to factor a trinomial by grouping.<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Factor a Trinomial in the Form ax^2+bx+c Using the Grouping Technique\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/agDaQ_cZnNc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n<h2>Factoring a Perfect Square Trinomial<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A perfect square trinomial is a trinomial that can be written as the square of a binomial<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a^2 + 2ab + b^2 = (a + b)^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a^2 - 2ab + b^2 = (a - b)^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identifying Perfect Square Trinomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">First and last terms are perfect squares<\/li>\n<li class=\"whitespace-normal break-words\">Middle term is twice the product of the square roots of the first and last terms<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Factoring Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the squares of [latex]a[\/latex] and [latex]b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Verify the middle term is [latex]2ab[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Write as [latex](a \\pm b)^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]16x^2 - 24x + 9[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q423709\">Show Answer<\/button><\/p>\n<div id=\"q423709\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Identify squares: [latex]\\begin{align*} 16x^2 &= (4x)^2 \\ 9 &= 3^2 \\end{align*}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Check middle term: [latex]\\begin{align*} -24x &= 2(4x)(3) \\ -24x &= -24x \\quad \\text{(verified)} \\end{align*}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Write factored form: [latex]\\begin{align*} 16x^2 - 24x + 9 &= (4x - 3)^2 \\end{align*}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Factoring a Difference of Squares<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A difference of squares is an expression of the form [latex]a^2 - b^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Form:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a^2 - b^2 = (a+b)(a-b)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identifying a Difference of Squares:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Two terms<\/li>\n<li class=\"whitespace-normal break-words\">Both terms are perfect squares<\/li>\n<li class=\"whitespace-normal break-words\">One term is subtracted from the other<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Factoring Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the squares of [latex]a[\/latex] and [latex]b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Write as \u00a0[latex]\\left(a+b\\right)\\left(a-b\\right)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]49y^2 - 64[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q106877\">Show Answer<\/button><\/p>\n<div id=\"q106877\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Identify squares: [latex]\\begin{align*} 49y^2 &= (7y)^2 \\ 64 &= 8^2 \\end{align*}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Apply the difference of squares formula: [latex]\\begin{align*} 49y^2 - 64 &= (7y)^2 - 8^2 \\ &= (7y+8)(7y-8) \\end{align*}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see another example of how to factor a difference of squares.<iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Li9IBp5HrFA?si=89OEOlaK8aD-GrdB\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n<h2>Factoring the Sum and Difference of Cubes<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Sum of Cubes Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a^3 + b^3 = (a + b)(a^2 - ab + b^2)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Difference of Cubes Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]a^3 - b^3 = (a - b)(a^2 + ab + b^2)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">SOAP Mnemonic:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>S<\/strong>ame sign in first factor<\/li>\n<li class=\"whitespace-normal break-words\"><strong>O<\/strong>pposite sign between terms in second factor<\/li>\n<li class=\"whitespace-normal break-words\"><strong>A<\/strong>lways positive for last term in second factor<\/li>\n<li class=\"whitespace-normal break-words\"><strong>P<\/strong>ositive sign between factors<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Recognition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify expressions in the form [latex]a^3 \\pm b^3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Confirm that both terms are perfect cubes<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Factor the sum of cubes [latex]216{a}^{3}+{b}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q496204\">Show Solution<\/button><\/p>\n<div id=\"q496204\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(6a+b\\right)\\left(36{a}^{2}-6ab+{b}^{2}\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Factor the difference of cubes: [latex]1,000{x}^{3}-1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q510077\">Show Solution<\/button><\/p>\n<div id=\"q510077\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(10x - 1\\right)\\left(100{x}^{2}+10x+1\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following two video examples we show more binomials that can be factored as a sum or difference of cubes.<iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tFSEpOB262M?si=6SMNIzv3tugfjC9x\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/J_0ctMrl5_0?si=KHKyUENKhs8_o94U\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n<h2>Factoring Expressions with Fractional or Negative Exponents<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">General Approach:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the Greatest Common Factor (GCF) with fractional or negative exponents<\/li>\n<li class=\"whitespace-normal break-words\">Factor out the GCF, including the variable with the lowest exponent<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Fractional Exponents:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Treat fractional exponents similarly to integer exponents<\/li>\n<li class=\"whitespace-normal break-words\">Factor out the variable with the lowest fractional exponent<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Negative Exponents:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Include negative exponents in the factoring process<\/li>\n<li class=\"whitespace-normal break-words\">Factor out the term with the most negative (or least positive) exponent<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Combining Terms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">After factoring, combine like terms within parentheses if possible<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Factor [latex]2{\\left(5a - 1\\right)}^{\\frac{3}{4}}+7a{\\left(5a - 1\\right)}^{-\\frac{1}{4}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q989124\">Show Solution<\/button><\/p>\n<div id=\"q989124\" class=\"hidden-answer\" style=\"display: none\">[latex]{\\left(5a - 1\\right)}^{-\\frac{1}{4}}\\left(17a - 2\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video provides more examples of how to factor expressions with negative exponents.<iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/4w99g0GZOCk?si=aUnt9qbNJbTpT6eq\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">The following video provides more examples of how to factor expressions with fractional exponents.<iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/R6BzjR2O4z8?si=PFDc9vazfllfRnnK\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":32,"module-header":"- Select Header 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