Introduction to Derivatives: Background You’ll Need 2

  • Break down different types of polynomial expressions into simpler parts using several factoring methods

Factor Polynomials

Factoring is central to simplifying expressions, solving equations, and understanding polynomial behavior. Factoring involves breaking down expressions into simpler, constituent parts. A key step in this process is identifying the greatest common factor (GCF), which simplifies polynomials by dividing out commonalities and reducing complexity.

Greatest Common Factor

The greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. 

[latex]4[/latex] is the GCF of [latex]16[/latex] and [latex]20[/latex] because it is the largest number that divides evenly into both [latex]16[/latex] and [latex]20[/latex].

The GCF of polynomials works the same way. The greatest common factor (GCF) of a polynomial is the largest polynomial that divides evenly into each term of the polynomial.

[latex]4x[/latex] is the GCF of [latex]16x[/latex] and [latex]20{x}^{2}[/latex] because it is the largest polynomial that divides evenly into both [latex]16x[/latex] and [latex]20{x}^{2}[/latex].

When factoring a polynomial expression, our first step is to check to see if each term contains a common factor. If so, we factor out the greatest amount we can from each term. 

To make it less challenging to find this GCF of the polynomial terms, first look for the GCF of the coefficients, and then look for the GCF of the variables.

To factor out a GCF from a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property “backwards” to rewrite the polynomial in a factored form.

The distributive property allows us to multiply a number by a sum or difference inside parentheses and add or subtract the results. Conversely, when we see a common factor shared by all terms, we can factor it out, effectively reversing the distributive process.

  • Using the distributive property: [latex]a\left(b+c\right)=ab+ac[/latex].
  • Factoring out a common factor: [latex]ab+ac=a\left(b+c\right)[/latex].

This principle shows us that multiplication distributed across a sum can be “undone” through factoring, revealing the GCF and the remaining terms of the polynomial.

How To: Given a Polynomial Expression, Factor Out the Greatest Common Factor

  1. Identify the GCF of the coefficients.
  2. Identify the GCF of the variables.
  3. Combine to find the GCF of the expression.
  4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.
  5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.

Factor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[/latex].

Factoring Quadratic Trinomials with a Leading Coefficient of [latex]1[/latex]

When factoring polynomials, starting with the greatest common factor (GCF) is standard. However, the GCF is not always the key to simplification, particularly for polynomials without a common factor. Let’s look at some examples.

The quadratic trinomial [latex]{x}^{2}+5x+6[/latex] has a GCF of [latex]1[/latex], but it can be written as the product of the factors [latex]\left(x+2\right)[/latex] and [latex]\left(x+3\right)[/latex].

The trinomial [latex]{x}^{2}+10x+16[/latex] can be factored using the numbers [latex]2[/latex] and [latex]8[/latex], because [latex]2 \times 8 =16[/latex] and [latex]2 + 8 = 10[/latex]. The trinomial can be rewritten as the product of [latex]\left(x+2\right)[/latex] and [latex]\left(x+8\right)[/latex].

To factor trinomials like [latex]{x}^{2}+bx+c[/latex], find two numbers that multiply to [latex]c[/latex] and add up to [latex]b[/latex].

It’s a common misconception that all trinomials can be broken down into binomial factors, but this isn’t always the case. While many polynomials can be factored in this way, revealing a product of simpler binomials, there are instances where a trinomial is prime and cannot be factored further using real numbers

How To: Factoring a Trinomial of the Form [latex]{x}^{2}+bx+c[/latex]

  1. Identify all the numbers that multiple together to get [latex]c[/latex].
  2. Of these numbers, find the pair of numbers where the sum equals [latex]b[/latex].
  3. Write the trinomial as the product of two binomials, [latex]\left(x+p\right)\left(x+q\right)[/latex].

Factor [latex]{x}^{2}+2x - 15[/latex].


Factoring by Grouping

When we have trinomials with leading coefficients other than 1, they can often be factored by grouping. This method involves breaking the middle term into two terms that can be factored separately, and then extracting the greatest common factor (GCF). 

The trinomial [latex]2{x}^{2}+5x+3[/latex] can be approached by writing it as [latex](2x^2+2x)+(3x+3)[/latex] and then factoring each group separately. This gives us [latex]2x(x+1)+3(x+1)[/latex]. We then factor out the common binomial [latex]\left(x+1\right)[/latex] to get the final factored form [latex](2x+3)(x+1)[/latex]

How To: Given a Trinomial in the Form [latex]a{x}^{2}+bx+c[/latex], Factor by Grouping

  1. Multiply [latex]a[/latex] and [latex]c[/latex] to find the key number.
  2. Find two numbers that multiply to the key number and add to [latex]b[/latex].
  3. Split the middle term, [latex]bx[/latex], using these two numbers and rewrite the trinomial.
  4. Group the terms into pairs and factor out the common factor from each group.
  5. Extract the common binomial factor from the groups.
  6. Write the original expression as the product of two binomials.

Factor [latex]5{x}^{2}+7x - 6[/latex] by grouping.


Factoring a Perfect Square Trinomial

A perfect square trinomial is one that can be expressed as a binomial squared. It occurs when you square a binomial, resulting in a trinomial where the first and last terms are perfect squares and the middle term is twice the product of the terms being squared.

[latex]\begin{array}{ccc}\hfill {a}^{2}+2ab+{b}^{2}& =& {\left(a+b\right)}^{2} \text{         When adding.} \hfill \\ & \text{and}& \\ \hfill {a}^{2}-2ab+{b}^{2}& =& {\left(a-b\right)}^{2} \text{        When subtracting.}\hfill \end{array}[/latex]
These formulas allow us to rewrite any perfect square trinomial in its factored form.

The trinomial [latex]49{x}^{2}-14x+1[/latex] factors into a binomial squared.

The first term [latex]49x^2[/latex] is the square of [latex]7x[/latex], and the last term [latex]1[/latex] is the square of [latex]1[/latex]. The middle term, [latex]−14x[/latex], is equal to twice the product of [latex]7x[/latex] and [latex]−1[/latex].

Therefore, the trinomial is a perfect square and its factored form is [latex]{\left(7x - 1\right)}^{2}[/latex].

How to: Factor a Perfect Square Trinomial:

  1. Check that both the first and last terms are perfect squares.
  2. Verify that the middle term is double the product of the square roots of the first and last terms.
  3. Express the trinomial as a squared binomial, [latex]{\left(a+b\right)}^{2}[/latex] or [latex]{\left(a-b\right)}^{2}[/latex], based on the sign of the middle term.

Factor [latex]25{x}^{2}+20x+4[/latex].


Factoring a Difference of Squares

A difference of squares occurs when you subtract one perfect square from another. It’s a special pattern in algebra where two square terms are separated by a minus sign, and it can be factored into two binomials with opposite signs.

[latex]{a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)[/latex]

This equation represents the factored form of a difference of squares.

Take the expression [latex]81y^2-100[/latex], for example.

Both [latex]81y^2[/latex] and [latex]100[/latex] are perfect squares, with [latex]81y^2[/latex] being [latex](9y)^2[/latex] and [latex]100[/latex] being [latex]10^2[/latex].

This expression can be factored into binomials as follows:

The factored form of [latex]81{y}^{2}-100[/latex] is [latex]\left(9y+10\right)\left(9y - 10\right)[/latex].

How To: Factor a Difference of Squares

  1. Identify the Squares: Start by ensuring both terms are perfect squares. In other words, each term can be written as some expression squared, such as [latex]a^2[/latex] and [latex]b^2[/latex].
  2. Determine the Roots: Find the square root of each term. The square root of [latex]a^2[/latex] is [latex]a[/latex], and the square root of [latex]b^2[/latex] is [latex]b[/latex].
  3. Set Up Binomials: Create two binomial expressions. One binomial will have a plus sign, and the other will have a minus sign between the terms.
  4. Write the Factored Form: Combine the binomials to form the factored expression: [latex]\left(a+b\right)\left(a-b\right)[/latex]

Factor [latex]9{x}^{2}-25[/latex].


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