Factoring Polynomials: Greatest Common Factor and Factoring by Grouping

Section 6.1: The Greatest Common Factor and Factoring by Grouping

Factoring Polynomials

Factoring is a fundamental process in algebra that involves rewriting a polynomial as a product of simpler expressions. This technique is essential for simplifying expressions, solving equations, and understanding polynomial structure.

• Definition: To factor a polynomial means to write it as a product of two or more expressions, rather than as a sum.

• Purpose: Factoring helps in solving equations, simplifying expressions, and finding roots of polynomials.

• Example: If we rewrite $x^2 + 5x + 6$ as $(x + 2)(x + 3)$, we have factored the polynomial.

Greatest Common Factor (GCF)

The greatest common factor (GCF) of a set of terms is the largest expression that divides each term exactly. Factoring out the GCF is often the first step in factoring polynomials.

• Definition: The GCF is the highest degree expression (including numbers and variables) that divides each term of the polynomial.

• Variable Part: The variable part of the GCF contains the smallest power of each variable present in all terms.

• Example: For the monomials $12x^2y$ and $18xy^2$, the GCF is $6xy$.

Finding the GCF: Step-by-Step

1. List the coefficients and variables for each term.

2. Find the largest number that divides all coefficients.

3. For each variable, use the lowest exponent present in all terms.

Examples

• Example 1a: Find the GCF of $8x^2$ and $12x$. Solution: The GCF is $4x$.

• Example 1b: Find the GCF of $15y^3$ and $25y^2$. Solution: The GCF is $5y^2$.

Factoring Out the GCF from a Polynomial

Once the GCF is identified, it can be factored out of the polynomial using the distributive property.

1. Determine the GCF of all terms.

2. Express each term as the product of the GCF and its other factor.

3. Use the distributive property to factor out the GCF.

• Example: Factor $28x + 35$. Step 1: GCF is $7$. Step 2: $28x = 7 \times 4x$, $35 = 7 \times 5$. Step 3: Factor out $7$: $28x + 35 = 7(4x + 5)$.

• Example: Factor $15xy + 25xy$. Step 1: GCF is $5xy$. Step 2: $15xy = 5xy \times 3$, $25xy = 5xy \times 5$. Step 3: Factor out $5xy$: $15xy + 25xy = 5xy(3 + 5)$.

Factoring Out the Negative of the GCF

Sometimes, it is preferable to factor out the negative of the GCF, especially when the leading coefficient is negative. This ensures the first term inside the parentheses is positive.

• Procedure: Factor out $-\text{GCF}$ instead of $\text{GCF}$.

• Example: Factor $-5x + 10$. Step 1: GCF is $5$. Step 2: Factor out $-5$: $-5x + 10 = -5(x - 2)$.

Factoring by Grouping

Factoring by grouping is a method used when a polynomial has four or more terms and no single GCF for all terms. Terms are grouped to find common factors within each group, then a common binomial factor is factored out.

1. Group terms with common monomial factors.

2. Factor out the common monomial factor from each group.

3. Factor out the remaining common binomial factor.

• Example: Factor $ax + ay + bx + by$ by grouping. Step 1: Group: $(ax + ay) + (bx + by)$ Step 2: Factor: $a(x + y) + b(x + y)$ Step 3: Factor out $(x + y)$: $(x + y)(a + b)$

• Example: Factor $x^2 + 2x + x + 2$ by grouping. Step 1: Group: $(x^2 + 2x) + (x + 2)$ Step 2: Factor: $x(x + 2) + 1(x + 2)$ Step 3: Factor out $(x + 2)$: $(x + 2)(x + 1)$

Factoring by Grouping with Negative Coefficients

Sometimes, a negative factor is used to obtain a common binomial factor for the two groupings.

• Example: Factor $ab - ac + cb - cc$ by grouping. Step 1: Group: $(ab - ac) + (cb - cc)$ Step 2: Factor: $a(b - c) + c(b - c)$ Step 3: Factor out $(b - c)$: $(b - c)(a + c)$

Summary Table: Factoring Methods

Method	When to Use	Steps	Example
Factoring out GCF	All terms share a common factor	Find GCF, rewrite terms, factor out GCF	$12x^2y + 18xy^2 = 6xy(2x + 3y)$
Factoring out negative GCF	Leading coefficient is negative	Find GCF, factor out negative GCF	$-8x + 16 = -8(x - 2)$
Factoring by grouping	No single GCF, four or more terms	Group terms, factor each group, factor common binomial	$ax + ay + bx + by = (x + y)(a + b)$

Additional info: Factoring is a foundational skill for solving polynomial equations and simplifying expressions in algebra. Mastery of GCF and grouping techniques is essential for progressing to more advanced factoring methods.