1. Algebra 1
Section 10.3

2. Factoring ax2 + bx + c (3x – 2)(x + 5) 3x2 + 15x – 2x – 10
Sum of the middle terms
First terms (FOIL)
Last terms (FOIL)

3. Find factors of +6 whose sum is +7.
Example 1
Factor 3x2 + 7x + 2.
ac = 6 and b = 7
Find factors of +6 whose sum is +7.
3x2 + 1x + 6x + 2

4. Example 1
3x2 + 1x + 6x + 2
Group the first two terms and the last two terms and factor the GCF from each grouping.
(3x2 + 1x) + (6x + 2)
x(3x + 1) + 2(3x + 1)
(x + 2)(3x + 1)

5. Factoring Trinomials of the Form ax2 + bx + c
Find factors of ac whose sum is b.
Rewrite the middle term as a sum of terms with these factors of ac as their coefficients.

6. Factoring Trinomials of the Form ax2 + bx + c
Factor the four-term polynomial by grouping the terms in pairs.
Factor the common monomial from each pair.
Factor the common binomial from each new term.

7. Factoring Trinomials of the Form ax2 + bx + c
Check your factorization by multiplying the factors.

8. Example 2 Factor 2x2 – 19x + 24. ac = 48 and b = -19
Find factors of +48 whose sum is -19.
Both factors must be negative.
2x2 – 3x – 16x + 24

9. Example 2 2x2 – 3x – 16x + 24 (2x2 – 3x) + (-16x + 24)

10. Find factors of -60 whose sum is -7.
Example 3
Factor 3x2 – 20 – 7x.
3x2 – 7x – 20
ac = -60 and b = -7
Find factors of -60 whose sum is -7.
3x2 – 12x + 5x – 20

11. Example 3 3x2 – 12x + 5x – 20 (3x2 – 12x) + (5x – 20)

12. Factoring
Remember: The first step in factoring a polynomial is to factor out any common monomial factors.
In Example 4, there is a common factor of 2x.

13. Find factors of 30 whose sum is -13.
Example 4
Factor 12x3 – 26x2 + 10x.
2x(6x2 – 13x + 5)
ac = 30 and b = -13
Find factors of 30 whose sum is -13.
2x(6x2 – 10x – 3x + 5)

14. Example 4 2x(6x2 – 10x – 3x + 5) 2x[(6x2 – 10x) + (-3x + 5)]

15. Example 5
Factor 2x2 + 3xy – 14y2.
Proceed as before, but remember the “y”.
ac = -28 and b = 3
Find factors of -28 whose sum is 3.
2x2 + 7xy – 4xy – 14y2

16. Example 5 2x2 + 7xy – 4xy – 14y2 (2x2 + 7xy) + (-4xy – 14y2)
x(2x + 7y) – 2y(2x + 7y)
(x – 2y)(2x + 7y)

17. Factoring ax2 + bx + c
It is also possible, for trinomials with small coefficients, to use a method of listing the combinations of coefficients.
Notice that Example 6 uses this method, and notice the similarities to Example 5.

18. Homework: pp