Factoring polynomials is a key skill in algebra. It's all about breaking down complex expressions into simpler parts. This process helps solve equations and simplify expressions, making math problems easier to handle.

The greatest common factor (GCF) and factoring by grouping are two important techniques. GCF works for polynomials with a common factor, while grouping is useful for longer expressions. Mastering these methods opens doors to solving trickier math problems.

Greatest Common Factor and Factoring by Grouping

Greatest common factor in polynomials

• Largest factor that divides all terms in a polynomial without leaving a remainder
• Found by determining the greatest common coefficient and variable factors among the terms
• Extracting the GCF involves factoring it out from each term in the polynomial
• Results in the GCF multiplied by the remaining terms in parentheses ($ax^2 + bx = x(ax + b)$)
• Simplifies the polynomial and is the first step in factoring completely
• Example: $12x^3 + 18x^2 = 6x^2(2x + 3)$, where $6x^2$ is the GCF
• Prime factorization can be used to find the GCF of numerical coefficients

Factor by grouping technique

• Technique used to factor polynomials with four or more terms and no apparent GCF
• Involves grouping terms into two or more groups that share a common factor
• Factor out the GCF from each group
• If the remaining expressions in parentheses are the same for all groups, factor out the common expression
• The final factored form will have the common expression multiplied by the factors from each group
• Helps to break down complex polynomials into simpler factors
• Example: $6x^2 + 3x - 4x - 2 = 3x(2x + 1) - 2(2x + 1) = (3x - 2)(2x + 1)$

Greatest common factor vs factor by grouping

• GCF method is used when the polynomial has an evident GCF that divides all terms ($ax + ay = a(x + y)$)
  • Suitable for polynomials with two or three terms (binomials and trinomials)
• Factor by grouping is used when the polynomial has four or more terms and no clear GCF
  • Terms are grouped to find common factors ($ax + ay + bx + by = (ax + bx) + (ay + by) = x(a + b) + y(a + b) = (a + b)(x + y)$)
• In some cases, a combination of both methods is necessary
  • Extract the GCF first, if one exists
  • Then apply factor by grouping to the remaining polynomial if it has four or more terms
• Recognizing the appropriate method saves time and simplifies the factoring process
• Examples:
  • GCF: $15x^2 - 25x = 5x(3x - 5)$
  • Factor by grouping: $2x^3 - 3x^2 + 4x - 6 = x(2x^2 - 3x) + 2(2x - 3) = (x + 2)(2x^2 - 3x)$

Factoring Basics

• Factoring is the process of breaking down a polynomial into simpler expressions
• Like terms are terms with the same variables raised to the same powers
• Grouping like terms can simplify the factoring process
• Binomials are expressions with two terms
• Trinomials are expressions with three terms