NAME
11-1
DATE
16. Geometry The area of a rectangle is
(6x2 7x 2) square inches. Find
binomial expressions to represent the
1, 2, 3, 16
Q
C
C
dimensions of this rectangle.
17. Standardized Test Practice Factor the trinomial v2 7v 12.
A (v 7)(v 5)
B (v 4)(v 3)
C (v 3)(v 4)
D (v 12)(v 5)
Factoring Differences of Squares (Pages 581–586)
You can use the difference of squares rule to factor binomials that can be
written in the form a2 b2. Sometimes the terms of a binomial have common
factors. If so, the GCF should always be factored out first.
Difference of Squares
b) or (a b)(a b)
a2 b2 (a b)(a Try These Together
Factor each polynomial, if
possible. If the polynomial
cannot be factored, write
prime.
EXAMPLES
A Factor b2 49.
B Factor 7g3h2 28g5.
1. x2 4
3.
b2 49
7g3h2 28g5
Check for a GCF.
(b)2 (7)2
b b b2 and 7 7 49
3
2
2
7g (h 4g )
GCF of 7g3h2 and 28g5 is 7g3.
(b 7)(b 7) Use the difference of squares.
7g3(h 2g)(h 2g)
h2 h h and 4g2 2g 2g.
2. y2 16
a2 144
HINT: Both terms of the binomial
must be squares. Also, the sum of
two squares cannot be factored
using the difference of two
squares rule.
PRACTICE
Factor each polynomial, if possible. If the polynomial cannot be factored,
write prime.
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
5. 4c2 7
6. 4z2 16
7. 9z2 19
8. 25 81x2
9. v2q2 0.49r2
10. a2b2 0.36c2
11. a2b2c2 x2y2z2
12. x2y2 3
13. t7 t3u4
14. x5 x3y2
15. 64k2 24
4 2
9 2
y . (Hint: Find fractions that when squared equal
16. Factor x 25
16
4
25
9
and .)
16
Answers: 1–9. For graphs, see Answer Key. 1. x 5; (5, 1); minimum 2. x 3; (3, 16); maximum
3. x 1; (1, 0); minimum 4. x 3; (3, 3); maximum 5. x 0.5; (0.5, 6.25); minimum 6. x 0; (0, 18); minimum
7. x 0; (0, 1); maximum 8. x 0; (0, 0); minimum 9. x 1; (1, 0); minimum 10. C
B
3.
4. 9b2 25
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Georgia Parent and Student Study Guide, Algebra 1
NAME
11-2
DATE
17. Standardized Test Practice Factor x2 (y
z)2.
1, 2, 3, 16
A (x y z)(x y z)
B (x y
Q
C
C
z)(x y z)
C (x y z)(x y z)
D (x y z)(x y z)
Perfect Squares and Factoring (Pages 587–593)
Products of the form (a b)2 and (a b)2 are called perfect squares, and
their expressions are called perfect square trinomials.
Perfect Square
Trinomials
(a b)2 a2 2ab b2
(a b)2 a2 2ab b2
Factoring a
Perfect Square
Trinomial
You can check whether a trinomial is a perfect square trinomial by checking that the
following conditions are satisfied.
• The first term is a perfect square.
• The third term is a perfect square.
• The middle term is either 2 or 2 times the product of the square root of the first
term and the square root of the second term.
EXAMPLE
Determine whether 4x2 4xy y2 is a perfect square trinomial.
If so, factor it.
Check the each of the following.
• Is the first term a perfect square? 4x 2 (2x) 2 yes
• Is the last term a perfect square? y 2 ( y) 2 yes
• Is the middle term twice the product of 2x and y? 4xy 2(2x)( y)
yes
4xy is a perfect square trinomial.
So,
4x2 4xy y2 (2x)2 2(2x)(y) (y)2
(2x y)2
4x2
y2
PRACTICE
Determine whether each trinomial is a perfect square trinomial. If so, factor
it. If the polynomial cannot be factored write prime.
1. m2 6m 9
2. x2 10x 25
3. t2 14t 49
4. x2 3x 4
5. y2 12y 36
6. k2 22k 121
Factor each polynomial. If the polynomial cannot be factored write prime.
7. x2 16x 64
B
4.
C
B
5.
C
B
A
7.
8.
10. 4m2 20m 25
11. 100h2 9
12. 4z3 16z2 16z
13. 3x2 24x 48
14. n2 1.8n 0.81
15. 7x2 5.6x 1.12
C
A
6.
9. x2 3x 9
B
A
1
16. Factor y2 4y 36. (Hint: Check to see if the trinomial is a perfect
9
Answers: 1. 5, 2 2. 4 3. 4. 3, 1 5. 4 6. 7. 1 8. between 1 and 2; between 1 and 2 9. 10. 2, 0
11. between 1 and 2; between 4 and 5 12. 2, 2 13. A
3.
8. 2q2 30q 8
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Georgia Parent and Student Study Guide, Algebra 1
NAME
DATE
square trinomial.)
11-3
1, 3
Q
C
C
17. Standardized Test Practice Factor the
trinomial 5a2 30a 45.
B 5(a 3)
A (5a 3)2
C (a 3)2
D 5(a 3)2
Solving Equations by Factoring (Pages 594–600)
You can use the zero product property to solve equations by factoring.
For all numbers a and b, if ab 0, then a 0, b 0, or both a and b equal 0.
Zero Product Property
EXAMPLES
A Solve x2 64 16x.
x 2 64 16x
16x 64 0
(x 8)(x 8) 0
x2
x80
x8
Rewrite the equation.
Factor the perfect square trinomial.
or x 8 0
x8
The solution set is {8}.
B
3.
C
C
A
B
5.
C
B
6.
A
7.
B
A
8.
19. $53; $65 20. D
10. no real roots 11. , 4 12. 1.5 13. 3, 9 14. 2.15, 0.15 15. 3.3, 0.3 16. no real roots 17. 0.3, 7.7 18. none
3
1
Answers: 1. 4, 2 2. 4, 8 3. no real roots 4. 4.24; 0.24 5. 3, 5 6. no real roots 7. 1 8. 3.73, 0.27 9. 1, 2
4.
© Glencoe/McGraw-Hill
87
Georgia Parent and Student Study Guide, Algebra 1
NAME
11-4
DATE
B Solve 12y3 11y2 15y
12y3 11y 2 15y 0
y(12y 2 11y 15) 0
y(4y 3)(3y 5) 0
1, 2, 3, 16
Q
C
C
Rewrite the equation.
Factor the GCF, y 2.
y 0 or 4y 3 0 or 3y 5 0
4y 3
3y 5
5
y
3
3
y 4
5
3
, 0, .
The solution set is 4
3
Try These Together
Solve each equation. Check each solution.
1. a2 9a 20 0
2. x2 16x
3. b2 10b
HINT: Remember that you may have more than one solution, so record your solutions as a solution set.
10. x2 11x 24 0
11x2 6x
g 2
PRACTICE
Solve each equation. Check
each solution.
4. y2 7y 12
10) 0
5)(2a 7) 0
5. (z 10)(z
6. (3a 7. z2 11z 24 0
18 0
21 0
8. k2 9k 9. x2 4x 11. 2x3 12. 5g 6 13. Geometry The triangle at the right has
an area of
63 square inches. Find the height h of the triangle.
h in.
(2h 4) in.
1
(Hint: Area of triangle bh)
2
14. Standardized Test Practice Solve the equation k(k 15)(k 8) 0.
A {15, 8}
B {15, 0, 8}
C {0, 8, 15}
D {8, 0, 15}
Chapter 10 Review
Rewind / Fast Forward
B
C
C
B
C
B
6.
A
7.
8.
B
A
“Rewind” by factoring each polynomial completely. Then cross off the answer
in the right column. “Fast forward” by multiplying your answer to check it.
2
A
5.
1
8. 2 9. 1 10. 4.
© Glencoe/McGraw-Hill
1
Answers: 1. 140.30 2. 0.05 3. 1.15 4. 8.89 5. 1.09 6–11. For graphs, see Answer Key. 6. 1 7. 8
11. 2 12. 12 13. 0.5 14. 5 15. C
3.
88
Georgia Parent and Student Study Guide, Algebra 1
NAME
11-5
DATE
The letters that are left will spell an outdated
technology.
Rewind
1, 2, 3
Q
C
C
1. 18x 9xy
2. 4x3 6x
3. x2 64
4. x2 16
5. 2x2 32
6. x2 6x 8
7. x2 6x 8
8. x2 x 12
9. x2 x 12
10. x2 2x xy 2y
11. xy 4y x2 4x
B
3.
12. 4x 8y x2 2xy
C
C
A
B
5.
C
B
6.
A
7.
B
A
8.
Fast Forward
(x 2)( y 1)
E
(x 3)(x 4)
N
(x 4)(x 3)
D
Answers: 1. decay 2. growth 3. decay 4a. $580.65 4b. $577.81 4c. $603.11 4d. $602.82 4e. $579.41 5. C
4.
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Georgia Parent and Student Study Guide, Algebra 1
NAME
11
(x 4)(x 2)
W
(x 4)(x 4)
I
(x 4)(x 4)
N
(x 8)(x 8)
G
(x 8)(x 8)
S
(x 2)(x 4)
P
(x 2)( y x)
H
(x 2)(x y)
A
(x 4y)(x 2)
T
(x 4)( y x)
I
(x 4)(x 2y)
O
2(2x2 3x)
T
2(x 4)(x 4)
S
y
O
DATE
6
4
2
x
y
18
15
12
9
6
3
O 1 2 3 4 5 6x
–2
–4
–6
–8
y
O 1 2 3 4 5 6x
6
5
4
3
2
1
O
y
1
2
x
Answers are located on page 113.
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Georgia Parent and Student Study Guide, Algebra 1