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8-4 Factoring ax2 + bx + c
Warm Up
Find each product.
1. (x – 2)(2x + 7)
2x2 + 3x – 14
2. (3y + 4)(2y + 9) 6y2 + 35y + 36
3. (3n – 5)(n – 7) 3n2 – 26n + 35
Find each trinomial.
4. x2 +4x – 32 (x – 4)(x + 8)
5. z2 + 15z + 36 (z + 3)(z + 12)
6. h2 – 17h + 72 (h – 8)(h – 9)
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Warm Up
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Objective
Factor quadratic trinomials of the form
ax2 + bx + c.
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Find a pair of numbers that multiply to the first number, and
add to the second.
1. Multiplies to 24; Adds to 10
6&4
2. Multiplies to -8; Adds to -2
-4 & 2
3. Multiplies to 15; Adds to 8
5&3
4. Multiplies to 21; Adds to -10
-7 & -3
5. Multiplies to -27; Adds to 6
9 & -3
6. Multiplies to -49; Adds to 0
7 & -7
Holt Algebra 1
7. Multiplies to 36; Adds to 13
9&4
8. Multiplies to -19; Adds to -18
-19 & 1
9. Multiplies to 6; Adds to -5
-3 & -2
10. Multiplies to -14; Adds to -5
-7 & 2
11. Multiplies to 16; Adds to 10
8&2
12. Multiplies to 15; Adds to 16
15 & 1
8-4 Factoring ax2 + bx + c
So, to factor a2 + bx + c, check the factors of a and
the factors of c in the binomials. The sum of the
products of the outer and inner terms should be b.
Product = c
Product = a
(
X+
)(
x+
) = ax2 + bx + c
Sum of outer and inner products = b
Holt Algebra 1
8-4 Factoring ax2 + bx + c
The guess and check method is usually not the
most efficient method of factoring a trinomial. Look
at the product of (x + 3) and (x + 4).
x2
12
(x + 3)(x +4) = x2 + 7x + 12
3x
4x
The coefficient of the middle term is the sum of 3
and 4. The third term is the product of 3 and 4.
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c When c is Positive
Factor each trinomial. Check your answer.
2x2 + 17x + 21
(
x+
)(
x+
a = 2 and c = 21,
) Outer + Inner = 17.
Two numbers that multiply to 42; Add to 17: 14 & 3
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: 2x
GCF of 2nd group: 3
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Remember!
When b is negative and c is positive, the factors
of c are both negative.
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
x2 + 6x + 5
(
x+
)(
x+
a = 1 and c = 5,
) Outer + Inner = 6.
Two numbers that multiply to 5; Add to 6: 5 & 1
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: x
GCF of 2nd group: 1
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
2x2 + 9x - 18
(
x+
)(
x+
a = 2 and c = -18,
) Outer + Inner = 9.
Two numbers that multiply to -36; Add to 9: 12 & -3
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: 2x
GCF of 2nd group: -3
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
3x2 - 16x + 16
(
x+
)(
x+
a = 3 and c = 16,
) Outer + Inner = -16.
Two numbers that multiply to 48; Add to -16: -12 & -4
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: 3x
GCF of 2nd group: -4
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
4x2 - 15x - 4
(
x+
)(
x+
a = 4 and c = -4,
) Outer + Inner = 13.
Two numbers that multiply to -16; Add to -15: -16 & 1
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: 4x
GCF of 2nd group: 1
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
x2 - 8x + 15
(
x+
)(
x+
a = 1 and c = 15,
) Outer + Inner = -8.
Two numbers that multiply to 15; Add to -8: -5 & -3
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: x
GCF of 2nd group: -3
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
6x2 + 17x + 5
(
x+
)(
x+
a = 6 and c = 5,
) Outer + Inner = 17.
Two numbers that multiply to 30; Add to 17: 15 & 2
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: 3x
GCF of 2nd group: 1
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
x2 - 5x + 6
(
x+
)(
x+
a = 1 and c = 6,
) Outer + Inner = -5.
Two numbers that multiply to 6; Add to -5: -2 & -3
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: x
GCF of 2nd group: -3
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
3x2 + 11x - 4
(
x+
)(
x+
a = 3 and c = -4,
) Outer + Inner = 13.
Two numbers that multiply to -12; Add to 11: 12 & -1
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: 3x
GCF of 2nd group: -1
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
9x2 - 15x + 4
(
x+
)(
x+
a = 9 and c = 4,
) Outer + Inner = -15.
Two numbers that multiply to 36; Add to -15: -12 & -3
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: 3x
GCF of 2nd group: -1
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
3x2 + 13x + 12
(
x+
)(
x+
a = 3 and c = 12,
) Outer + Inner = 13.
Two numbers that multiply to 36; Add to 13: 9 & 4
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: 3x
GCF of 2nd group: 4
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor x2 + 10x + 21. Show that the original polynomial and the factored form have the
same value for n = 0, 1, 2, 3, and 4.
x2 + 10x + 21
a = 1 and c = 21,
( x + )( x + ) Outer + Inner = 10.
Two numbers that multiply to 21; Add to 10: 7 & 3
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: x
GCF of 2nd group: 3
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 4A Continued
Evaluate the original polynomial and the
factored form for n = 0, 1, 2, 3, and 4.
y2 + 10y + 21
y
(y + 7)(y + 3)
y
0
(0 + 7)(0 + 3) = 21
0
02 + 10(0) + 21 = 21
1
(1 + 7)(1 + 3) = 32
1
12 + 10(1) + 21 = 32
2
(2 + 7)(2 + 3) = 45
2
22 + 10(2) + 21 = 45
3
(3 + 7)(3 + 3) = 60
3
32 + 10(3) + 21 = 60
4
(4 + 7)(4 + 3) = 77
4
42 + 10(4) + 21 = 77
The original polynomial and the factored form
have the same value for the given values of n.
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
x2 - 9
(
x+
)(
x+
a = 1 and c = -9,
) Outer + Inner = 0.
Two numbers that multiply to -9; Add to 0: -3 & 3
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: x
GCF of 2nd group: 3
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
x2 - 36
(
x+
)(
x+
a = 1 and c = -36,
) Outer + Inner = 0.
Two numbers that multiply to -36; Add to 0: -6 & 6
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: x
GCF of 2nd group: 6
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
x2 - 81
(
x+
)(
x+
a = 1 and c = -81,
) Outer + Inner = 0.
Two numbers that multiply to -81; Add to 0: -9 & 9
Break up the b term into the
two numbers that you found.
Factor by grouping; GCF of 1st group: x
GCF of 2nd group: 9
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
x2 + 25
Two numbers that multiply to 25; Add to 0: …..?
THE KEY to this problem is having c be a negative number, not a positive
Holt Algebra 1
8-4 Factoring ax2 + bx + c
When the leading coefficient is negative,
factor out –1 from each term before using
other factoring methods.
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Caution
When you factor out –1 in an early step, you
must carry it through the rest of the steps.
Holt Algebra 1
8-4 Factoring ax2 + bx + c
Factor. Check your answer.
Holt Algebra 1