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You factored trinomials of the form x2 + bx + c.
• LEQ: How do we factor trinomials of the form
ax2 + bx + c & solve equations of the form
ax2 + bx + c = 0?
• prime polynomial
Factor ax2 + bx + c
A. Factor 5x2 + 27x + 10.
In this trinomial, a = 5, b = 27, and c = 10. You need to
find two numbers with a sum of 27 and with a product of
5 ● 10 or 50. Make an organized list of the factors of 50
and look for the pair of factors with the sum of 27.
Factors of 50
Sum of Factors
1, 50
51
2, 25
27
The correct factors
are 2 and 25.
5x2 + 27x + 10 = 5x2 + mx + px + 10 Write the pattern.
= 5x2 + 2x + 25x + 10 m = 2 and p = 25
Factor ax2 + bx + c
= (5x2 + 2x) + (25x + 10)
= x(5x + 2) + 5(5x + 2)
= (x + 5)(5x + 2)
Answer:
Group terms with
common factors.
Factor the GCF.
Distributive
Property
Factor ax2 + bx + c
= (5x2 + 2x) + (25x + 10)
= x(5x + 2) + 5(5x + 2)
= (x + 5)(5x + 2)
Group terms with
common factors.
Factor the GCF.
Distributive
Property
Answer: (x + 5)(5x + 2) or (5x + 2)(x + 5)
Factor ax2 + bx + c
B. Factor 4x2 + 24x + 32.
The GCF of the terms 4x2, 24x, and 32 is 4. Factor this
term first.
4x2 + 24x + 32 = 4(x2 + 6x + 8)
Distributive
Property
Now factor x2 + 6x + 8. Since the lead coefficient is 1,
find the two factors of 8 whose sum is 6.
Factors of 8
1, 8
2, 4
Sum of Factors
9
6
The correct factors
are 2 and 4.
Factor ax2 + bx + c
Answer:
Factor ax2 + bx + c
Answer: So, x2 + 6x + 4 = (x + 2)(x + 4). Thus, the
complete factorization of 4x2 + 24x + 32 is
4(x + 2)(x + 4).
A. Factor 3x2 + 26x + 35.
A. (3x + 7)(x + 5)
B. (3x + 1)(x + 35)
C. (3x + 5)(x + 7)
D. (x + 1)(3x + 7)
A. Factor 3x2 + 26x + 35.
A. (3x + 7)(x + 5)
B. (3x + 1)(x + 35)
C. (3x + 5)(x + 7)
D. (x + 1)(3x + 7)
B. Factor 2x2 + 14x + 20.
A. (2x + 4)(x + 5)
B. (x + 2)(2x + 10)
C. 2(x2 + 7x + 10)
D. 2(x + 2)(x + 5)
B. Factor 2x2 + 14x + 20.
A. (2x + 4)(x + 5)
B. (x + 2)(2x + 10)
C. 2(x2 + 7x + 10)
D. 2(x + 2)(x + 5)
Factor ax2 – bx + c
Factor 24x2 – 22x + 3.
In this trinomial, a = 24, b = –22, and c = 3. Since b is
negative, m + p is negative. Since c is positive, mp is
positive. So m and p must both be negative. Therefore,
make a list of the negative factors of 24 ● 3 or 72, and
look for the pair of factors with the sum of –22.
Factors of 72 Sum of Factors
–1, –72
–73
–2, –36
–38
–3, –24
–27
–4, –18
–22
The correct factors
are –4 and –18.
Factor ax2 – bx + c
24x2 – 22x + 3 = 24x2 + mx + px + 3 Write the pattern.
= 24x2 – 4x – 18x + 3
m = –4 and p = –18
= (24x2 – 4x) + (–18x + 3) Group terms with
common factors.
= 4x(6x – 1) + (–3)(6x – 1) Factor the GCF.
= (4x – 3)(6x – 1)
Answer:
Distributive
Property
Factor ax2 – bx + c
24x2 – 22x + 3 = 24x2 + mx + px + 3 Write the pattern.
= 24x2 – 4x – 18x + 3
m = –4 and p = –18
= (24x2 – 4x) + (–18x + 3) Group terms with
common factors.
= 4x(6x – 1) + (–3)(6x – 1) Factor the GCF.
= (4x – 3)(6x – 1)
Answer: (4x – 3)(6x – 1)
Distributive
Property
Factor 10x2 – 23x + 12.
A. (2x + 3)(5x + 4)
B. (2x – 3)(5x – 4)
C. (2x + 6)(5x – 2)
D. (2x – 6)(5x – 2)
Factor 10x2 – 23x + 12.
A. (2x + 3)(5x + 4)
B. (2x – 3)(5x – 4)
C. (2x + 6)(5x – 2)
D. (2x – 6)(5x – 2)
Determine Whether a Polynomial is Prime
Factor 3x2 + 7x – 5, if possible.
In this trinomial, a = 3, b = 7, and c = –5. Since b is
positive, m + p is positive. Since c is negative, mp is
negative, so either m or p is negative, but not both.
Therefore, make a list of all the factors of 3(–5) or –15,
where one factor in each pair is negative. Look for the
pair of factors with a sum of 7.
Factors of –15
–1, 15
1, –15
–3,
Sum of Factors
14
–14
5
2
3, –5
–2
Determine Whether a Polynomial is Prime
There are no factors whose sum is 7. Therefore,
3x2 + 7x – 5 cannot be factored using integers.
Answer:
Determine Whether a Polynomial is Prime
There are no factors whose sum is 7. Therefore,
3x2 + 7x – 5 cannot be factored using integers.
Answer: 3x2 + 7x – 5 is a prime polynomial.
Factor 3x2 – 5x + 3, if possible.
A. (3x + 1)(x – 3)
B. (3x – 3)(x – 1)
C. (3x – 1)(x – 3)
D.
prime
Factor 3x2 – 5x + 3, if possible.
A. (3x + 1)(x – 3)
B. (3x – 3)(x – 1)
C. (3x – 1)(x – 3)
D.
prime
Solve Equations by Factoring
ROCKETS Mr. Nguyen’s science class built a model
rocket for a competition. When they launched their
rocket outside the classroom, the rocket cleared the
top of a 60-foot high pole and then landed in a nearby
tree. If the launch pad was 2 feet above the ground,
the initial velocity of the rocket was 64 feet per
second, and the rocket landed 30 feet above the
ground, how long was the rocket in flight? Use the
equation h = –16t2 + vt + h0.
h = –16t2 + vt + h0
Equation for height
30 = –16t2 + 64t + 2
0 = –16t2 + 64t – 28
h = 30, v = 64, h0 = 2
Subtract 30 from each
side.
Solve Equations by Factoring
0 = –4(4t2 – 16t + 7)
Factor out –4.
0 = 4t2 – 16t + 7
Divide each side by –4.
0 = (2t – 7)(2t – 1)
Factor 4t2 – 16t + 7.
2t – 7 = 0 or 2t – 1 = 0
2t = 7
2t = 1
Zero Product Property
Solve each equation.
Divide.
Solve Equations by Factoring
again on its way down. Thus, the rocket was in flight for
about 3.5 seconds before landing.
Answer:
Solve Equations by Factoring
again on its way down. Thus, the rocket was in flight for
about 3.5 seconds before landing.
Answer: about 3.5 seconds
When Mario jumps over a hurdle, his feet leave the
ground traveling at an initial upward velocity of
12 feet per second. Find the time t in seconds it takes
for Mario’s feet to reach the ground again. Use the
equation h = –16t2 + vt + h0.
A. 1 second
B. 0 seconds
C.
D.
When Mario jumps over a hurdle, his feet leave the
ground traveling at an initial upward velocity of
12 feet per second. Find the time t in seconds it takes
for Mario’s feet to reach the ground again. Use the
equation h = –16t2 + vt + h0.
A. 1 second
B. 0 seconds
C.
D.