7.1 Greatest Common Factor and Factor by Grouping

Greatest Common Factor and Factoring

Factoring is the reverse of multiplying. Instead of expanding expressions, you're breaking them down into simpler pieces that multiply together. The greatest common factor (GCF) is the starting point for almost all factoring problems, so getting comfortable with it now will pay off throughout the rest of this unit.

Greatest Common Factor in Expressions

The GCF of an algebraic expression is the largest factor that divides every term evenly, with no remainder. It includes both the largest shared number and the highest shared power of each variable.

To find the GCF of an expression:

1. Break each term into its prime factors (include variables).
2. Identify the factors that appear in every term.
3. For numbers, take the largest shared factor. For variables, take the lowest power that appears across all terms.

That third step trips people up. You take the lowest power of each variable, not the highest, because the GCF has to divide into every term.

Examples:

• $6x^2 + 9x$
  • $6x^2 = 2 \cdot 3 \cdot x \cdot x$ and $9x = 3 \cdot 3 \cdot x$
  • Shared factors: $3$ and $x$
  • GCF: $3x$
• $12x^3y^2 + 8x^2y$
  • $12x^3y^2 = 2^2 \cdot 3 \cdot x^3 \cdot y^2$ and $8x^2y = 2^3 \cdot x^2 \cdot y$
  • Shared factors: $2^2 = 4$, $x^2$ (lowest power of $x$), and $y$ (lowest power of $y$)
  • GCF: $4x^2y$

Factoring Polynomials with the GCF

Once you've found the GCF, you divide each term by it and write the expression as a product.

Steps:

1. Find the GCF of all terms.
2. Divide each term by the GCF.
3. Write the result as: GCF $\times$ (quotient of each term).

Example: Factor $15x^3 + 25x^2$

1. GCF of $15x^3$ and $25x^2$ is $5x^2$.
2. $15x^3 \div 5x^2 = 3x$ and $25x^2 \div 5x^2 = 5$.
3. Factored form: $5x^2(3x + 5)$

You can always check your answer by distributing the GCF back through the parentheses. If you get the original expression, you factored correctly.

Factor by Grouping

Factoring by grouping is used when a polynomial has four terms and no single GCF works across all of them. The idea is to split the expression into two pairs, factor each pair separately, and then pull out the binomial they share.

Steps:

1. Group the four terms into two pairs.
2. Factor the GCF out of each pair.
3. Check that both groups now contain the same binomial factor.
4. Factor out that common binomial.

If the binomials don't match after step 2, try rearranging the terms into different pairs.

Example: Factor $6x^3 + 9x^2 - 4x - 6$

1. Group: $(6x^3 + 9x^2) + (-4x - 6)$

2. Factor each group: $3x^2(2x + 3) - 2(2x + 3)$

3. Both groups contain $(2x + 3)$.

4. Factor it out: $(2x + 3)(3x^2 - 2)$

Notice in step 2 that the second group needed a $-2$ factored out (not $+2$) so the binomial would match. Watch your signs carefully here; sign errors are the most common mistake with grouping.

Key Vocabulary

• Like terms: Terms with the same variables raised to the same powers (e.g., $3x^2$ and $-7x^2$ are like terms).
• Coefficient: The numerical part of a term. In $-5x^3$, the coefficient is $-5$.
• Factored form: An expression written as a product of its factors, like $5x^2(3x + 5)$.