Training Integer Exponents and Factoring Factoring — GCF and Grouping
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Factoring — GCF and Grouping

22 min Integer Exponents and Factoring

Factoring Polynomials

Factoring is the reverse of multiplication — it takes a single expression and breaks it into a product of simpler pieces. Factoring is crucial because it lets you solve equations, simplify rational expressions, and analyze the behavior of functions.

The greatest common factor (GCF) is the largest expression that divides every term, and pulling it out is always the first step. When a polynomial has four terms, factoring by grouping — splitting the terms into two pairs and factoring each pair separately — often reveals a common binomial factor.

This lesson covers GCF factoring and grouping, the two most fundamental techniques that underpin all the specialized methods in the lessons ahead.

Factoring reverses multiplication — it rewrites a polynomial as a product of simpler expressions.

Step 1 — Always

Factor out the GCF first. The greatest common factor is the largest expression dividing every term.

Example 1

Factor $6x^3 + 9x^2 - 15x$.

GCF $= 3x$. $\;3x(2x^2 + 3x - 5)$.

Example 2

Factor $12a^2b^3 - 18a^3b^2 + 6a^2b^2$.

GCF $= 6a^2b^2$. $\;6a^2b^2(2b - 3a + 1)$.

Factoring by Grouping

Steps
  1. Group into two pairs.
  2. Factor the GCF from each pair.
  3. Factor out the common binomial.
Example 3

Factor $x^3 + 3x^2 + 2x + 6$.

  1. $(x^3 + 3x^2) + (2x + 6)$
  2. $x^2(x + 3) + 2(x + 3)$
  3. $(x + 3)(x^2 + 2)$
Example 4

Factor $2xy + 6x - 3y - 9$.

  1. $(2xy + 6x) + (-3y - 9)$
  2. $2x(y + 3) - 3(y + 3)$
  3. $(y + 3)(2x - 3)$

Practice Problems

1. $8x^2 + 12x$
2. $15a^3b - 10a^2b^2$
3. $x^3 - 2x^2 + 5x - 10$
4. $3xy + 9x - 2y - 6$
5. $21x^4 - 14x^3 + 7x^2$
6. $x^3 + x^2 + x + 1$
7. $4m^3n - 8m^2n^2 + 2mn$
8. $6a - 6b + a^2 - ab$
9. $18x^3y^2 + 27x^2y^3$
10. $x^3 - 4x^2 - 3x + 12$
Show Answer Key

1. $4x(2x + 3)$

2. $5a^2b(3a - 2b)$

3. $(x - 2)(x^2 + 5)$

4. $(y + 3)(3x - 2)$

5. $7x^2(3x^2 - 2x + 1)$

6. $(x + 1)(x^2 + 1)$

7. $2mn(2m^2 - 4mn + 1)$

8. $(a - b)(6 + a)$

9. $9x^2y^2(2x + 3y)$

10. $(x - 4)(x^2 - 3)$

Scientific Notation Factoring Trinomials