1. Introduction
Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product. We can factor certain expressions by writing them as the product of factors.
The Zero Product Property states that if the product of two factors is 0, then at least one of the factors is 0. After setting a quadratic equation equal to 0, we can sometimes factor the quadratic expression and solve the equation by setting each factor equal to 0.
3.2.2: Factoring

2. Key Concepts
The greatest common factor, or GCF, is the largest factor that two or more terms share.
You should always check to see if the terms of an expression have a greatest common factor before attempting to factor further.
The value of a for a quadratic expression in the form ax2 + bx + c is called the leading coefficient, or lead coefficient, because it is the coefficient of the term with the highest power.
3.2.2: Factoring

3. Key Concepts, continued
To factor a trinomial with a leading coefficient of 1 in the form x2 + bx + c, find two numbers d and e that have a product of c and a sum of b.
The factored form of the expression will be (x + d)(x + e).
When finding d and e, be careful with the signs.
The table that follows shows what the signs of d and e will be based on the signs of b and c.
3.2.2: Factoring

4. Key Concepts, continued Signs of b, c, d, and e+ –
Opposite signs; the number with the larger absolute value is positive.
Opposite signs; the number with the larger absolute value is negative.
3.2.2: Factoring

5. Key Concepts, continued
You may be able to factor expressions with lead coefficients other than 1 in your head or by using guess-and-check.
Expressions with lead coefficients other than 1 in the form ax2 + bx + c can sometimes be factored by grouping.
If you struggle to factor expressions in your head or by using guess-and-check, factoring by grouping is a more structured alternative.
3.2.2: Factoring

6. Key Concepts, continued
Factoring by Grouping
Begin by finding two numbers d and e whose product is ac and whose sum is b.
Rewrite the expression by replacing bx with dx + ex: ax2 + dx + ex + c.
Factor the greatest common factor from ax2 + dx.
Factor the greatest common factor from ex + c.
Factor the greatest common factor from the resulting expression.
3.2.2: Factoring

7. Key Concepts, continued
A quadratic expression in the form (ax)2 – b2 is called a difference of squares.
The difference of squares (ax)2 – b2 can be written in factored form as (ax + b)(ax – b).
Some expressions cannot be factored. These expressions are said to be prime.
Although the difference of squares is factorable, the sum of squares is prime.
For example, (3x)2 – 52 = 9x2 – 25 = (3x + 5)(3x – 5), but (3x) = 9x2 + 30x + 25 and is not factorable.
3.2.2: Factoring

8. Common Errors/Misconceptions
forgetting to consider the signs of a, b, and c
treating a lead coefficient other than 1 as if it were a 1
multiplying the terms of an expression given in factored form to solve an equation
solving an expression that is not part of an equation
confusing the difference of squares with the sum of squares
3.2.2: Factoring

9. Guided Practice Example 2 Solve 8x2 – 8 = –x2 + 56 by factoring.

10. Guided Practice: Example 2, continued
Rewrite the equation so that all terms are on one side.
8x2 – 8 = –x2 + 56
Original equation
9x2 – 8 = 56
Add x2 to both sides.
9x2 – 64 = 0
Subtract 56 from both sides.
3.2.2: Factoring

11. Guided Practice: Example 2, continued
Factor the difference of squares.
The expression on the left side can be rewritten in the form (3x)2 – 82.
We can use this form to rewrite the expression as the difference of squares to factor the expression.
(3x + 8)(3x – 8) = 0
3.2.2: Factoring

12. ✔ Guided Practice: Example 2, continued
Use the Zero Product Property to solve.
The expression will equal 0 only when one of the factors is equal to 0.
Set each factor equal to 0 and solve.
8x2 – 8 = –x when ✔
3.2.2: Factoring

13. Guided Practice: Example 2, continued
3.2.2: Factoring

14. Guided Practice Example 3 Solve x2 + 8x = 20 by factoring.

15. Guided Practice: Example 3, continued
Rewrite the equation so that all terms are on one side of the equation.
x2 + 8x = 20
Original equation
x2 + 8x – 20 = 0
Subtract 20 from both sides.
3.2.2: Factoring

16. Guided Practice: Example 3, continued Find the factors.
The lead coefficient of the expression is 1, so begin by finding two numbers whose product is –20 and whose sum is 8.
The numbers are –2 and 10 because (–2)(10) = –20 and – = 8.
Therefore, the factors are (x – 2) and (x + 10).
3.2.2: Factoring

17. Guided Practice: Example 3, continued
Write the expression as the product of its factors.
(x – 2)(x + 10) = 0
3.2.2: Factoring

18. ✔ Guided Practice: Example 3, continued
Use the Zero Product Property to solve.
The expression will equal 0 only when one of the factors is equal to 0. Set each factor equal to 0 and solve.
x – 2 = 0 x + 10 = 0
x = 2 x = –10
x2 + 8x = 20 when x = 2 or x = –10. ✔
3.2.2: Factoring

19. Guided Practice: Example 3, continued
3.2.2: Factoring