The Main Idea

• Definition of Factoring:
  • Breaking down a polynomial into simpler parts
  • Products of these parts equal the original polynomial
• Greatest Common Factor (GCF):
  • Largest factor common to all terms in a polynomial
  • Includes both numerical coefficients and variables
• Steps for Factoring using GCF:
  • Identify GCF of numerical coefficients
  • Identify GCF of variables (lowest exponent for each variable)
  • Combine numerical and variable GCFs
  • Factor out the combined GCF
• Distributive Property in Reverse:
  • [latex]ab + ac = a(b + c)[/latex]
  • Used to factor out the GCF
• Importance of Factoring:
  • Simplifies expressions
  • Aids in solving equations
  • Helps in understanding algebraic structures

The Main Idea

• Standard Form:
  • The trinomial is in the form [latex]x^2 + bx + c[/latex]
  • This factors into [latex](x + p)(x + q)[/latex]
• Coefficient Relationships:
  • The sum of p and q equals b: [latex]p + q = b[/latex]
  • The product of p and q equals c: [latex]pq = c[/latex]
• Factor Identification Process:
  • Find two numbers that multiply to give c
  • These same two numbers should add to give b
  • These numbers become p and q in the factored form
• Sign Considerations:
  • If c is positive, p and q have the same sign as b
  • If c is negative, p and q have opposite signs, with the larger magnitude having the same sign as b
• Verification:
  • The factored form [latex](x + p)(x + q)[/latex] should expand back to [latex]x^2 + bx + c[/latex]

The Main Idea 

• Applicable Trinomials:
  • Used for trinomials in the form [latex]ax^2 + bx + c[/latex] where [latex]a \neq 1[/latex]
• Coefficient Relationship:
  • Need to find two numbers m and n such that [latex]m \times n = a \times c[/latex] and [latex]m + n = b[/latex]
• Rewriting the Trinomial:
  • Express the middle term bx as the sum of two terms: [latex]mx + nx[/latex]
  • Rewrite the trinomial as a four-term polynomial: [latex]ax^2 + mx + nx + c[/latex]
• Grouping Process:
  • Group the first two and last two terms: [latex](ax^2 + mx) + (nx + c)[/latex]
  • Factor out the GCF from each group
• Common Binomial Extraction:
  • After grouping and factoring, a common binomial factor should emerge
  • Factor out this common binomial to complete the factorization
• Final Factored Form:
  • The result will be the product of two binomials: [latex](px + q)(rx + s)[/latex]
  • Where [latex]pr = a[/latex] and [latex]qs = c[/latex]
• Verification:
  • Multiplying out [latex](px + q)(rx + s)[/latex] should yield the original trinomial [latex]ax^2 + bx + c[/latex]

The Main Idea

• Definition:
  • A perfect square trinomial is a trinomial that can be written as the square of a binomial
• General Forms:
  • [latex]a^2 + 2ab + b^2 = (a + b)^2[/latex]
  • [latex]a^2 - 2ab + b^2 = (a - b)^2[/latex]
• Identifying Perfect Square Trinomials:
  • First and last terms are perfect squares
  • Middle term is twice the product of the square roots of the first and last terms
• Factoring Process:
  • Identify the squares of [latex]a[/latex] and [latex]b[/latex]
  • Verify the middle term is [latex]2ab[/latex]
  • Write as [latex](a \pm b)^2[/latex]

The Main Idea

• Definition:
  • A difference of squares is an expression of the form [latex]a^2 - b^2[/latex]
• General Form:
  • [latex]a^2 - b^2 = (a+b)(a-b)[/latex]
• Identifying a Difference of Squares:
  • Two terms
  • Both terms are perfect squares
  • One term is subtracted from the other
• Factoring Process:
  • Identify the squares of [latex]a[/latex] and [latex]b[/latex]
  • Write as  [latex]\left(a+b\right)\left(a-b\right)[/latex]

The Main Idea

• Sum of Cubes Formula:
  • [latex]a^3 + b^3 = (a + b)(a^2 - ab + b^2)[/latex]
• Difference of Cubes Formula:
  • [latex]a^3 - b^3 = (a - b)(a^2 + ab + b^2)[/latex]
• SOAP Mnemonic:
  • Same sign in first factor
  • Opposite sign between terms in second factor
  • Always positive for last term in second factor
  • Positive sign between factors
• Recognition:
  • Identify expressions in the form [latex]a^3 \pm b^3[/latex]
  • Confirm that both terms are perfect cubes

The Main Idea

• General Approach:
  • Identify the Greatest Common Factor (GCF) with fractional or negative exponents
  • Factor out the GCF, including the variable with the lowest exponent
• Fractional Exponents:
  • Treat fractional exponents similarly to integer exponents
  • Factor out the variable with the lowest fractional exponent
• Negative Exponents:
  • Include negative exponents in the factoring process
  • Factor out the term with the most negative (or least positive) exponent
• Combining Terms:
  • After factoring, combine like terms within parentheses if possible