Download 8-4 Factoring ax2 + bx + c

8-4 Factoring ax2 + bx + c
Warm Up
Find each product.
1. (x – 2)(2x + 7)
2. (3y + 4)(2y + 9)
3. (3n – 5)(n – 7)
Find each trinomial.
4. x2 +4x – 32
5. z2 + 15z + 36
6. h2 – 17h + 72
8-4 Factoring ax2 + bx + c
Learning Goals
2,3. Students will factor quadratic
trinomials of the form
ax2 + bx + c
ax2 - bx + c
ax2 + bx – c
ax2 - bx - c
4. Students will factor quadratic trinomials
with a negative a value
8-4 Factoring ax2 + bx + c
To factor a trinomial like ax2 + bx + c into its
binomial factors, write two sets of parentheses
( x + )( x + ).
Write two numbers that are factors of a next to the
x’s and two numbers that are factors of c in the
other blanks. Multiply the binomials to see if you are
correct.
(3x + 2)(2x + 5) = 6x2 + 19x + 10
8-4 Factoring ax2 + bx + c
Example 2,3: Factoring
ax2 + bx + c;
ax2 - bx + c;
ax2 + bx - c;
ax2 – bx - c
Factor using the ac method. Check your answer.
A. 6x2 + 11x + 4
B. 3x2 – 16x + 16
C. 3n2 + 11n – 4
D. 3x2 – 2x – 8
8-4 Factoring ax2 + bx + c
Example 2,3: Factoring
ax2 + bx + c;
ax2 - bx + c;
ax2 + bx - c;
ax2 – bx - c
Factor using the ac method. Check your answer.
A. 6x2 + 11x + 4
8-4 Factoring ax2 + bx + c
Example 2,3: Factoring
ax2 + bx + c;
ax2 - bx + c;
ax2 + bx - c;
ax2 – bx - c
Factor using the ac method. Check your answer.
B. 3x2 – 16x + 16
8-4 Factoring ax2 + bx + c
Example 2,3: Factoring
ax2 + bx + c;
ax2 - bx + c;
ax2 + bx - c;
ax2 – bx - c
Factor using the ac method. Check your answer.
C. 3n2 + 11n – 4
8-4 Factoring ax2 + bx + c
Example 2,3: Factoring
ax2 + bx + c;
ax2 - bx + c;
ax2 + bx - c;
ax2 – bx - c
Factor using the ac method. Check your answer.
D. 3x2 – 2x – 8
8-4 Factoring ax2 + bx + c
Example 2,3: Factoring
ax2 + bx + c;
ax2 - bx + c;
ax2 + bx - c;
ax2 – bx - c
Factor using the ac method. Check your answer.
a. 6x2 + 11x + 3
b. 9x2 – 15x + 4
c. 6x2 + 7x – 3
d. 4x2 – 15x – 4
e. 2x2 + 17x + 21
f. 3x2 - 13x + 12
g. 2x2 + 9x – 18
h. 4n2 – n – 3
i. 6x2 + 17x + 5
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
8-4 Factoring ax2 + bx + c
Since you need to check all the factors of a and the
factors of c, it may be helpful to make a table. Then
check the products of the outer and inner terms to
see if the sum is b. You can multiply the binomials
to check your answer.
Product = c
Product = a
(
X+
)(
x+
) = ax2 + bx + c
Sum of outer and inner products = b
8-4 Factoring ax2 + bx + c
Remember!
When b is negative and c is positive, the factors
of c are both negative.
8-4 Factoring ax2 + bx + c
When c is negative, one factor of c will be
positive and the other factor will be negative.
Only some of the factors are shown in the
examples, but you may need to check all of the
possibilities.
8-4 Factoring ax2 + bx + c
When the leading coefficient is negative,
factor out –1 from each term before using
other factoring methods.
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.
8-4 Factoring ax2 + bx + c
Example 4: Factoring ax2 + bx + c When a is Negative
Factor using the ac method. Check your answer.
A. –2x2 – 5x – 3.
B. –6x2 – 17x – 12
8-4 Factoring ax2 + bx + c
Example 4: Factoring ax2 + bx + c When a is Negative
Factor using the ac method. Check your answer.
a. –3x2 – 17x – 10
b. –3x2 + 2x + 8
8-4 Factoring ax2 + bx + c
Lesson Quiz
Factor each trinomial. Check your answer.
1. 5x2 + 17x + 6
(5x + 2)(x + 3)
2. 2x2 + 5x – 12
(2x– 3)(x + 4)
3. 6x2 – 23x + 7
(3x – 1)(2x – 7)
4. –4x2 + 11x + 20 (–x + 4)(4x + 5)
5. –2x2 + 7x – 3
(–2x + 1)(x – 3)
6. 8x2 + 27x + 9
(8x + 3)(x + 3)