NAME
12-1
DATE
Factor each polynomial.
5. 7b2 42b
10xz2 30z6
8. 8s3 24s2q
6. 15m2n 27mn2
9. 16g 14gh2
11. 6y3 21y2 4y 14
7.
10. 36k5 24k3 18k
12. 3x3 x2 6x 2
13. 4w3 3wz 8w2 6z
14. Geometry The area of a rectangle is represented by 10x3 15x2 4x 6.
Its dimensions are represented by binomials in x that have prime number
coefficients. What are the dimensions of the rectangle?
15. Standardized Test Practice Factor the polynomial 4wf 8w.
A 4(wf 2)
B 4w(f 2)
C 4w(f 8)
D w(4f 8)
Factoring Trinomials (Pages 574–580)
Use the guess and check strategy and the FOIL method to factor a trinomial.
1
1
EXAMPLE
Factor 22x 6x2 8.
First, rewrite the trinomial so that the terms are in descending order. Then check for a GCF.
22x 6x2 8 6x2 22x 8
2(3x2 11x 4)
The GCF of the terms is 2. Use the distributive property.
Now factor 3x2 11x 4.

↓
↓
3x2 11x 4
The product of 3
and 4 is 12.
2
3x ( ? ? ) x 4
↑↑

You need to find two integers whose
product is 12 and whose sum is 11.
Factors of 12
Sum of Factors
3, 4
3, 4
1, 12
1, 12
3 4 1
3 4 1
1 12 11
1 (12) 11
no
no
no
yes
Stop listing factors when you find a pair that works.
B
3.
C
C
A
B
5.
C
B
6.
A
7.
B
A
8.
y1
6
x2
13. , x 4 14. , y 6, 1 15. D
y1
x2
2x
x1
4
a2
4x
10 5x
b2
3
1
7. , x 2 8. , x 0 9. x 5, x 5 10. , b 2 11. , x 1, 1 12. , a 7, 2
2
x5
5
3
a3
9yz
xy
a2
x2
1
8
Answers: 1. , a 0 2. , x 1, 5 3. , x 0 4. x 0, y 0 5. x, x 0, y 0, z 0 6. , a 0, 3
x2
4.
© Glencoe/McGraw-Hill
91
MS Parent and Student Study Guide, Algebra 1
NAME
12-2
DATE
3x2 11x 4 3x2 [1 (12)]x 4
Select the factors 1 and 12.
3x2 1x 12x 4
(3x2 1x) (12x 4)
monomial factor.
Group terms that have a common
x(3x 1) 4(3x 1)
(x 4)(3x 1)
Factor.
Use the distributive property.
Therefore, 6x2 22x 8 2(x 4)(3x 1).
PRACTICE
Complete.
1. b2 b 6 (b 3)(b ? )
2. a2 2a 8 (a ? )(a 2)
3. x2 3x 10 (x ? )(x 2)
4. k2 9k 18 (k 6)(k ? )
1 x
1
5. 8g2 4g 12 ( ? 4)(2g 3)
1
6. 5n2 22n 8 (5n ? )(n 4)
2 1
Factor each trinomial.
7. x2 x 12
8. y2 5y 14
9. k2 15k 50
10. a2 4a 12
11. z2 11z 24
12. 3s2 9s 30
13. 2x2 3x 20
14. 9x2 18x 5
15. 20x2 17x 3
16. Geometry The area of a rectangle is (6x2 7x 2) square inches. Find
binomial expressions to represent the 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
a2 b2 (a b)(a b) or (a b)(a b)
EXAMPLES
7g3h2 28g5
Check for a GCF.
7g3(h2 4g2)
GCF of 7g3h2 and 28g5 is 7g3.
7g3(h 2g)(h 2g) h2 h h and 4g2 2g 2g.
2
ab
Answers: 1. x8
4
2. 2y
x 3z 2
3. 10ab 4. c
16a b
5. 2
2
x3
17. 2
25n
6. 4
18. C
5x
2
7. 14 8. 4a 9. 92
x x2
x5
4x
10. 3x2 7x 6 11. x4
x 16
12. 2
© Glencoe/McGraw-Hill
x1
16. 2
b2 49
(b)2 (7)2
b b b2 and 7 7 49
(b 7)(b 7) Use the difference of squares.
y3
C
B
A
y6
8.
15. A
7.
2
C
B
B
6.
x4
C
A
5.
x 3x
14. 4.
x 6x 9
B
3.
B Factor 7g3h2 28g5.
6
13. 2
A Factor b2 49.
MS Parent and Student Study Guide, Algebra 1
NAME
12-3
DATE
Try These Together
Factor each polynomial, if possible. If the
polynomial cannot be factored, write
prime.
1. x2 4
2. y2 16
3. 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. 1
4. 9b2 25
1
7. 9z2 19
5. 4c2 7
6. 4z2 16
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
17. Standardized Test Practice Factor x2 (y z)2.
A (x y z)(x y z)
B (x y 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.
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 third term.
C
C
C
B
A
8.
EXAMPLE
yz
2
3. a
40bc
4. x3
x8
5. 4 6. 4
b5
7. 2 8. 93
a4
17. 4 18. A
2
1
9. n2 1 10. 5b3 11. k 5 12. 8 13. © Glencoe/McGraw-Hill
3
2. A
7.
m
B
B
6.
2
Answers: 1. 3
A
5.
x3
4.
(a b)2 a2 2ab b2
(a b)2 a2 2ab b2
1
14. 2x2 8 15. n2 7n 12 16. B
3.
Perfect Square
Trinomials
MS Parent and Student Study Guide, Algebra 1
NAME
12-4
DATE
Determine whether 4x2 4xy y2 is a perfect
square trinomial.
If so, factor it.
Check 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
So, 4x2 4xy y2 is a perfect square trinomial.
4. x2 3x 4
36
121
4x2 4xy y2 (2x)2 2(2x)(y) (y)2
(2x y)2
PRACTICE
Factor each polynomial. If the
polynomial cannot be
factored write prime.
Determine whether each
trinomial is a perfect square
trinomial. If so, factor it. If the
polynomial cannot be
factored write prime.
7. x2 16x 64
8
9
1. m2 6m 9
25
49
2. x2 10x
3. t2 14t 13. 3x2 24x 48
5. y2 12y
6. k2 22k
8. 2q2 30q
9. x2 3x 11. 100h2 9
10. 4m2 20m 25
3
2
12.
4z 16z 16z
14. n2 1.8n 0.81
15. 7x2 5.6x 1.12
1
16. Factor y2 4y 36. (Hint: Check to see if the trinomial is a perfect
9
square trinomial.)
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
Rewrite the equation.
Factor the perfect square trinomial.
2
x
x2
4
Answers: 1. 2
n3
3
k
k
2
2. y 5 3. 4
x1
2
13. x 4 2x 3
5
14. 3x 5 y1
1
15. y2 5y 7 x 1
7
4. x 5 5. x 2 6. a 4 7. 2y 2 8. x 7 94
3
12. 2n b2
6
9. 2b 1 © Glencoe/McGraw-Hill
t
C
B
A
4
8.
8
A
7.
3
2
11. t C
B
B
6.
x3
C
A
5.
6
10. 2x 3 4.
16. x2 x 2 17. D
B
3.
MS Parent and Student Study Guide, Algebra 1
NAME
DATE
x80
x8
12-5
or x 8 0
x8
O
B
J
E
C
T
I
V
E
S
The solution set is {8}.
8b
B Solve 12y3 11y2 15y
12y3 11y 2 15y 0 Rewrite the equation.
y(12y 2 11y 15) 0 Factor the GCF, y 2.
y(4y 3)(3y 5) 0
y 0 or 4y 3 0 or 3y 5 0
4y 3
3y 5
The solution set is 3
,
4
0,
5
3
b2 10b
3.
5
y
3
3
y 4
HINT: Remember that you may have more than one
solution, so record your solutions as a solution set.
.
Try These Together
PRACTICE
Solve each equation.
Check each solution.
Solve each equation. Check
each solution.
1. a2 9a 20 0
2. x2 16x
10) 0
4. y2 7y 12
5. (z 10)(z
6. (3a 5)(2a 7) 0
7. z2 11z 24 0
8. k2 9k 18 0
10. x2 11x 24 0
9. x2 4x 21 0
11. 2x3 11x2 6x
12. 5g 6 g 2
13. Geometry The triangle at the right has an area of
63 square inches. Find the height h of the triangle.
1
(Hint: Area of triangle bh)
2
h in.
(2h 4) in.
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
B
3.
C
C
A
B
5.
C
B
6.
A
7.
B
A
8.
Rewind / Fast Forward
“Rewind” by factoring each polynomial completely. Then cross off the answer
in the right column. “Fast forward” by multiplying your answer to check it.
The letters that are left will spell an outdated technology.
16. B
m
1
Answers: 1. 3
x
2. x
6
3. 2
1
4. 2
5y 3
5. k2
k
6. 2x 7. 1 8. 95
x2
2n 3
2n 6
9. 1 10. a4
8
11. 2d 1 12. 5
1
13. 14. 1
© Glencoe/McGraw-Hill
x 3
15. 4.
MS Parent and Student Study Guide, Algebra 1
NAME
DATE
Rewind
12-6
1. 18x 9xy
2. 4x3 6x
3. x2 64
4. x2 16
5. 2x2 32
9. x2 x 12
6. x2 6x 8
10. x2 2x xy 2y
7. x2 6x 8
11. xy 4y x2 4x
8. x2 x 12
12. 4x 8y x2 2xy
Fast Forward
B
D
(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
Answers: 1.
2
2.
15
7x
3.
13. D
5x 10
xy2
a
7a 9
4. 3
2
3x 6
17
5. 2x 8
3
6. 96
3
10x
4x 4
9x
7. x 9x 18
8x 33
8. 2
x 9
5x 11x 12
9. 2
2
x 16
9x 8
10. 2
© Glencoe/McGraw-Hill
x 5x 6
B
A
4x 23x
12. 2
C
x 100
A
8.
(x 4)(x 3)
C
B
B
7.
N
C
A
5.
6.
(x 3)(x 4)
2
4.
E
x 10x 3
11. 2
3.
(x 2)( y 1)
MS Parent and Student Study Guide, Algebra 1
NAME
12-7
DATE
(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
2(x 8)(x 8)
R
2(x2 16)
A
2x(2x2 3)
O
3x(6 3y)
C
9x(2 y)
E
9x(2 xy)
K
Graphing Quadratic Functions (Pages 611–617)
Quadratic
Function
1
Axis of
Symmetry
A quadratic function is a function that can be written in the form f(x ) ax2 bx c
where a 0. The graph of a quadratic function is a parabola.
a is positive: parabola opens upward and vertex is a minimum point of the function
a is negative: parabola opens downward and vertex is a maximum point of the function
Parabolas have symmetry, which means that when they are folded in half on a line that
passes through the vertex, each half matches the other exactly. This line is called the
axis of symmetry.
b
1
Axis of symmetry for graph of y ax2 bx c , where a 0, is x .
2a
EXAMPLE
Given the equation y x2 2x 3, find the equation for the axis of
symmetry, the coordinates of the vertex, and graph the equation.
In the equation y x2 2x 3, a 1 and b 2.
symmetry.
Substitute these values into the equation for the axis of
b
axis of symmetry: x 2a
2
x or 1
2(1)
Since you know the line of symmetry, you know the
x-coordinate for the vertex is 1.
B
C
C
A
B
5.
C
B
A
B
A
Coordinates of vertex: (x, y) (1, 2)
Graph the vertex and the line of symmetry, x 1.
x5
3x 14
4. 2
b
5. 4
y
x
6. 4
3
97
n1
m7
m 5
7. 5
2t
t 2t 3
8. 2
9. B
© Glencoe/McGraw-Hill
8n 13
3. 8.
x7
7.
y x 2 2x 3
y 1 2 3 or 2 Replace x with 1.
4x 26
2. 6.
x
4.
x2 4
Answers: 1. 3.
MS Parent and Student Study Guide, Algebra 1
NAME
12-8
DATE
Using the equation, you can find another point on the graph. The point (3, 6) is 2 units right of
the axis of symmetry. Since the graph is symmetrical, if you go
2 units left of the axis and 6 units up, you will find a third point on the graph, (1, 6). Repeat this
for several other points. Then sketch the parabola.
y
(–1, 6)
(3, 6)
(1, 2)
minimum point
x
O
x=1
PRACTICE
Write the equation of the axis of symmetry and find the coordinates of the
vertex of the graph of each equation. State if the vertex is a maximum or
minimum. Then graph the equation.
1. y x2 10x 24
2. y x2 6x 7
3. y x2 2x 1
4. y 3x2 18x 24
5. y x2 x 6
6. y 2x2 18
7. y x2 1
8. y 3x2
9. y x2 2x 1
10. Standardized Test Practice What is the vertex of the graph of
y 1 4x 2x2?
A (2, 1)
C (1, 1)
B (2, 17)
D (1, 7)
Solving Quadratic Equations by Graphing (Pages
620–627)
The solutions of a quadratic equation are called the roots of the equation.
You can find the real number roots by finding the x-intercepts or zeros of the
related quadratic function. Quadratic equations can have two distinct real
roots, one distinct root, or no real roots. These roots can be found by
graphing the equation to see where the parabola crosses the x-axis.
EXAMPLES
Describe the real roots of the quadratic equations whose related functions
are graphed below.
B
3.
C
C
A
B
5.
C
B
6.
A
7.
B
A
8.
13. 4, 4 14. no solution 15. 10, 4 16. C
4. 3 5. 7, 1 6. 0, 4 7. 4 8. 1 9. no solution 10. 7, 2 11. 1, 2 12. 3
98
1
2
© Glencoe/McGraw-Hill
Answers: 1. 14 2. 1, 5 3. 3, 4.
MS Parent and Student Study Guide, Algebra 1
NAME
DATE
12
y
A
y
B
y
C
x
O
x
O
O
The parabola crosses the
x-axis twice. One root is
between 1 and 2, and the
other is between 4 and 5.
x
Since the vertex of the
parabola lies on the x-axis
the function has one
distinct root, 2.
Connect the
This parabola does not intersect
answers
to real
each
the x-axis, so
there are no
problem
in
roots. The solution set is .the
following order:
PRACTICE
Connect #1 to #2.
State the real roots of each quadratic equation whose related function is
Connect #3 to #4.
graphed below.
Connect #5 to #6.
Connect #2 to #7.
Connect #5 to #3.
Connect #7 to #8.
8
3
35x
Connect #4 to #6.
x2
x
x2
2x 1
x2
2
x5
3
x2 3
x2 9
2
35x
x2
2x 8
x2 2x 8
2x2 8
x
3x
xy
x4
2x 4
x2
22
35x
3x
y
x
3
1
x2 3x 1
x3
x2 2x 1
x
3y
2x 1
x2 2x 1
x2 2x 1
6
Answers are located on page 113.
© Glencoe/McGraw-Hill
99
MS Parent and Student Study Guide, Algebra 1