1. Algebra 1
Section 10.2

2. product of the constants
Factoring x2 + bx + c
(x + 5)(x + 7) = x2 + 12x + 35
constants
sum of the constants
product of the constants

3. Factoring x2 + bx + c Write the factors of the first term.
Find factors of the constant term, c, whose sum is b. Write these factors as the last terms in the binomials.

4. Factoring x2 + bx + c
Check the results by multiplying.

5. Example 1
Factor x2 – 5x + 6.
3. Write these factors as the last terms in the binomials.
4. Check by multiplying.
Find two factors of +6 whose sum is -5.
1. Write the factors of x2 as the first terms in the binomials.
-2 and -3
(x – 2)(x – 3)
(x )(x )
x2 – 2x – 3x + 6
x2 – 5x + 6

6. Find two factors of -6 whose sum is +5.
Example 2
Factor y2 + 5y – 6.
Find two factors of -6 whose sum is +5.
6 and -1
(y + 6)(y – 1)
(y )(y )

7. Determining the Signs of the Factors of c
If the sign of c is positive,
a. the signs are the same and
b. the factors will have the sign of the trinomial’s middle term.

8. Determining the Signs of the Factors of c
If the sign of c is negative,
a. the signs are different and
b. the larger factor will have the sign of the trinomial’s middle term.

9. Find two factors of -72 whose sum is +1.
Example 3
Factor x2 + x – 72.
Find two factors of -72 whose sum is +1.
9 and -8
(x + 9)(x – 8)
(x )(x )

10. Example 4 Factor x2 + 3x + 6. Find two factors of 6 whose sum is +3.
There are no such factors.
Therefore, this trinomial is prime.
(x )(x )

11. Factoring
The first step in factoring a polynomial is to factor out any common monomial factors.
When the constant term is a large number, a more organized approach may be helpful.

12. Example 5
Factor 2a2 – 64a – 288.
First, factor out the common factor of 2.
2(a2 – 32a – 144)
Then, find two factors of that have a sum of -32.

13. Example 5
2(a2 – 32a – 144)
Two factors of -144 that have a sum of -32 are +4 and -36.
2(a + 4)(a – 36)
It is still wise to check your answer by multiplying!

14. Factoring Some trinomials contain two variables.
When factoring a trinomial of the form x2 + bxy + cy2, be sure to include a factor of y in the second term of each binomial.

15. Find two factors of +24 whose sum is +11.
Example 6
Factor x2 + 11xy + 24y2.
Find two factors of +24 whose sum is +11.
3 and 8
(x + 3y)(x + 8y)
(x y)(x y)

16. Homework: pp