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Name: ________________________ Class: ___________________ Date: __________
ID: A
Algebra 2 Ch. 8.1
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
____
1. Distance varies directly as time because as time increases, the distance traveled increases proportionally. The
speed of sound in air is about 335 feet per second. How long would it take for sound to travel 11,725 feet?
a. 35 sec
c. 30 sec
b. 45 sec
d. 40 sec
2. Given: y varies inversely as x, and y = 4 when x = –4. Write and graph the inverse variation function.
4
a. y = −x
c. y = x
b.
1
y = −x
d.
1
y=−
16
x
Name: ________________________
____
3. Determine whether the data set represents a direct variation, an inverse variation, or neither.
x
2
3
4
a.
b.
____
b.
c.
d.
____
____
y
420
280
210
Direct variation
Neither
c.
Inverse variation
10 − x 2 − 3x
. Identify any x-values for which the expression is undefined.
x 2 + 2x − 8
−x − 5
x+4
The expression is undefined at x = −4.
−x − 5
x+4
The expression is undefined at x = 2 and x = −4.
x+5
x+4
The expression is undefined at x = 2 and x = −4.
x+5
x+4
The expression is undefined at x = −4.
4. Simplify
a.
____
ID: A
5. Multiply
8x 4 y 2 9xy 2 z 6
⋅
. Assume that all expressions are defined.
3z 3
4y 4
a.
6x 4 yz 2
c.
6x 5 z 3
b.
6x 5 y 8 z 9
d.
3
2
6. Divide
5x 3
25
÷ 9 . Assume that all expressions are defined.
2
3x y 3y
a.
xy 8
5
c.
b.
125x
9y 10
d.
x
5y 8
5
xy 8
x 2 + x − 30
= 11. Check your answer.
x−5
x=5
x = 16
x = −6
There is no solution because the original equation is undefined at x = 5.
7. Solve
a.
b.
c.
d.
x3 y2 z
2
Name: ________________________
____
8. Add
a.
b.
____
x+6
−12x − 59
+ 2
.
x − 7 x − 3x − 28
−11x − 53
x 2 − 2x − 35
x 2 + 10x + 24
(x + 4)(x − 7)
ID: A
c.
x+6
(x − 7)(x + 4)
d.
x+5
x+4
2x 2 − 48 x + 6
−
. Identify any x-values for which the expression is undefined.
x 2 − 16 x + 4
x−6
x−4
The expression is undefined at x = 4 and x = −4.
x 2 + 2x − 72
(x − 4) (x + 4)
The expression is undefined at x = 4 and x = −4.
x+6
x−4
The expression is undefined at x = 4 and x = −4.
x−6
x+4
The expression is undefined at x = 4 and x = −4.
9. Subtract
a.
b.
c.
d.
3
Name: ________________________
ID: A
Graph the function.
____ 10. y =
4
x
a.
c.
b.
d.
Simplify the rational expression.
x 2 − 2x − 24
x 2 − 5x − 36
x−6
a.
x+9
2
x + 4x − 21
____ 12.
x 2 + x − 42
x−3
a.
x−6
____ 11.
b.
x+6
x+9
c.
x+6
x−9
d.
x−6
x−9
b.
x+3
x+6
c.
x+7
x+6
d.
x−3
x+7
4
Name: ________________________
ID: A
Multiply.
____ 13.
q+5
4q
⋅
2
q+4
2
4q + 20q
a.
2
2
q + 20q
b.
2q
c.
d.
x 2 − 16
7x
⋅
6x
x+4
7(x + 4)
a.
6
7(x − 4)
b.
6
y 2 − 9 −5y
⋅
____ 15.
−2y
y+3
−5(y + 3)
a.
−2
4q 2 + 20q
2q + 8
4q + 20q 2
4q + 8
____ 14.
c.
d.
(x + 4) 2 (x − 4)
42x 2
(x − 4) 2 (x + 4)
42x 2
b.
(y + 3)
−2
c.
y−3
−2
d.
−5(y − 3)
−2
b.
x+5
x−5
c.
x+5
x−4
d.
9x + 4
5
b.
s−5
s
c.
s
s−5
d.
s−2
s 2 − 5s
b.
x
6
c.
−x − x
−24x
d.
1
6
b.
1
x−7
c.
x+7
x
d.
1
x+7
Divide.
x 2 + 9x + 20
÷
x 2 − 25
x−4
a.
x−5
2
s − 2s
÷
____ 17. 2
s + 3s − 10
s−2
a.
s−5
____ 16.
x+4
x−4
s−5
s+5
Add or subtract.
−x + 6 −x − 6
+
−12x
−12x
1
a.
3
x
7
−
____ 19. 2
x − 49 x 2 − 49
7
a.
x+7
____ 18.
5
Name: ________________________
____ 20.
ID: A
2x + 3 x − 5
−
x−4
x+2
x+8
a.
−6
x 2 + 16x − 14
b.
(x + 2)(x − 4)
c.
d.
____ 21. Factor the trinomial a 2 + 14a + 48.
a. (a + 14) (a + 1)
b. (a + 1) (a + 48)
c.
d.
x+8
(x + 2)(x − 4)
x 2 − 2x − 2
(x + 2)(x − 4)
(a + 6) (a + 8)
(a − 8) (a − 6)
____ 22. Factor 2x 2 + 7x + 6.
a. (x + 3) (2x + 2)
c. (x + 2) (x + 3)
b. (x + 2) (2x − 3)
d. (x + 2) (2x + 3)
____ 23. Let x 1 = 15, y 1 = 8, and y 2 = 5. Let y vary inversely as x. Find x 2 .
a. x 2 = 115
c. x 2 = 9.38
b. x 2 = 2.67
d. x 2 = 24
____ 24. Simplify the rational expression
3
x−5
1
b.
x−2
____ 25. Multiply. Simplify your answer.
a.
−
x2 − x − 6
x2 + x
⋅
.
2x 2 − 6x x 2 + 4x + 4
x+1
a.
2x + 4
x
b.
2
x +4
____ 26. Divide. Simplify your answer.
1 m− 8
÷
m
8m
8
a.
m−8
1(8m)
b.
m(m − 8)
____ 27. Divide.
m2 + 10m + 24
m+4
a. m + 6
b. m − 4
x−3
.
x − 5x + 6
2
c.
x−2
d.
x
x−2
c.
d.
c.
d.
c.
d.
6
2x 2 − 6
3x 2 − 2x + 4
1
16
8
m
m−8
8
m− 6
m+ 4
ID: A
Algebra 2 Ch. 8.1
Answer Section
MULTIPLE CHOICE
1. ANS: A
r = 335 ft per sec
Find the constant of variation r.
d = 335t
Write the direct variation function.
11,725 = 335t
Substitute.
t = 35
Solve.
It would take 35 seconds for sound to travel 11,725 feet.
Feedback
A
B
C
D
Correct!
Use the direct variation equation d = rt.
Set up a proportion and solve.
First, write the direct variation function. Then, substitute the given values and solve.
PTS: 1
DIF: Basic
REF: Page 570
OBJ: 8-1.2 Solving Direct Variation Problems
STA: 2A.10.G
TOP: 8-1 Variation Functions
1
NAT: 12.5.4.c
ID: A
2. ANS: D
k
y=x
4=
y varies inversely as x.
k
−4
Substitute the given values.
Solve for k.
–16 = k
y=−
16
x
Write the variation function.
16
Make a table of values to graph y = − x . Use both positive and negative values.
x
–8
–6
–4
0
4
6
8
y
2
2.6
4
Undefined
–4
−2.6
–2
Feedback
A
B
C
D
Inverse variation equations are in the form y = k/x.
This equation does not have the correct constant of variation. To find k, use xy = x and
substitute the given x- and y-values.
This equation does not have the correct constant of variation. To find k, use xy = k and
substitute the given x- and y-values.
Correct!
PTS:
OBJ:
STA:
KEY:
1
DIF: Average
REF: Page 571
8-1.4 Writing and Graphing Inverse Variation
2A.10.G
TOP: 8-1 Variation Functions
inverse variation | relationship | graph
2
NAT: 12.5.1.e
ID: A
3. ANS: C
In inverse variation, the product of the two quantities is constant.
x
2
3
4
y
420
280
210
xy
840
840
840
In direct variation, the ratio of the two quantities is constant.
x
y
x
y
2
420
1
210
3
280
3
280
4
210
2
105
This data set represents an inverse variation.
Feedback
A
B
C
If the ratio of each x – y pair is the same, the relationship is a direct variation.
If the product of each x – y pair is the same, the relationship is an inverse variation.
Correct!
PTS:
OBJ:
STA:
4. ANS:
−1(x 2
1
DIF: Basic
REF: Page 572
8-1.6 Identifying Direct and Inverse Variation
NAT: 12.5.1.e
2A.10.G
TOP: 8-1 Variation Functions
KEY: inverse variation | relationship
B
+ 3x − 10)
Factor −1 from the numerator and reorder the terms.
2
x + 2x − 8
−1(x + 5)(x − 2)
=
Factor the numerator and denominator.
(x + 4)(x − 2)
−x − 5
=
Divide the common factors and simplify.
x+4
The expression is undefined at those x-values, 2 and −4, that make the original denominator 0.
Feedback
A
B
C
D
Look at the original expression to find the values that make it undefined.
Correct!
Don't forget to redistribute the –1.
Don't forget to redistribute the –1. Look at the original expression to find the values that
make it undefined.
PTS: 1
NAT: 12.5.3.c
DIF: Average
STA: 2A.2.A
REF: Page 578
OBJ: 8-2.2 Simplifying by Factoring -1
TOP: 8-2 Multiplying and Dividing Rational Expressions
3
ID: A
5. ANS: C
Arrange the expressions so like terms are together:
8 ⋅ 9(x 4 ⋅ x)(y 2 ⋅ y 2 )z 6
3 ⋅ 4 ⋅ z3y4
.
Multiply the numerators and denominators, remembering to add exponents when multiplying:
Divide, remembering to subtract exponents: 6x 5 y 0 z 3 .
Since y 0 = 1, this expression simplifies to 6x 5 z 3 .
Feedback
A
B
C
D
A variable raised to the 0 power simplifies to 1.
When dividing powers with the same base, subtract the exponents.
Correct!
Multiply, then simplify.
PTS:
OBJ:
STA:
6. ANS:
1
DIF: Basic
REF: Page 578
8-2.3 Multiplying Rational Expressions
NAT: 12.5.3.c
2A.2.A
TOP: 8-2 Multiplying and Dividing Rational Expressions
A
5x 3
25
÷ 9
2
3x y 3y
=
9
5x 3 3y
⋅
3x 2 y 25
=
xy 8
5
Rewrite as multiplication by the reciprocal.
Simplify by canceling common factors.
Feedback
A
B
C
D
Correct!
To divide by a fraction, you multiply by its reciprocal.
To divide by a fraction, you multiply by its reciprocal.
Multiply the first fraction by the reciprocal of the second fraction.
PTS: 1
DIF: Basic
REF: Page 579
OBJ: 8-2.4 Dividing Rational Expressions
NAT: 12.5.3.c
STA: 2A.2.A
TOP: 8-2 Multiplying and Dividing Rational Expressions
4
72x 5 y 4 z 6
12z 3 y 4
.
ID: A
7. ANS: D
x 2 + x − 30
Note that x ≠ 5.
= 11
x−5
(x − 5) (x + 6)
Factor.
= 11
x−5
x + 6 = 11
The factor (x − 5) cancels.
x=5
Because the left side of the original equation is undefined when x = 5, there is no solution.
Feedback
A
B
C
D
Is the original equation defined for this value of x?
Factor the numerator and cancel common factors before solving for x. Is the original
equation defined for this value of x?
Factor the numerator and cancel common factors before solving for x. Is the original
equation defined for this value of x?
Correct!
PTS: 1
DIF: Average
REF: Page 579
OBJ: 8-2.5 Solving Simple Rational Equations
NAT: 12.5.3.c
STA: 2A.10.D
TOP: 8-2 Multiplying and Dividing Rational Expressions
8. ANS: D
x+6
−12x − 59
+
Factor the denominators. The LCD is (x + 4)(x − 7).
x − 7 (x + 4)(x − 7)
ÊÁ x + 4 ˆ˜ x + 6
ÊÁ x + 4 ˆ˜
−12x − 59
˜˜
ÁÁ
˜
+
= ÁÁÁÁ
Multiply
by
˜
ÁÁ x + 4 ˜˜˜ .
˜
Ë x + 4 ¯ x − 7 (x + 4)(x − 7)
Ë
¯
=
x 2 + 10x + 24
−12x − 59
+
(x + 4)(x − 7) (x + 4)(x − 7)
x 2 − 2x − 35
(x + 4)(x − 7)
(x + 5)(x − 7)
=
(x + 4)(x − 7)
x+5
=
x+4
=
Add the numerators.
Factor the numerator.
Divide the common factor.
Feedback
A
B
C
D
Find a common denominator before adding the fractions.
This is the first fraction rewritten with the common denominator. Add this to the
second fraction.
Find a common denominator before adding the fractions.
Correct!
PTS: 1
NAT: 12.5.3.c
DIF: Average
STA: 2A.2.A
REF: Page 584
OBJ: 8-3.3 Adding Rational Expressions
TOP: 8-3 Adding and Subtracting Rational Expressions
5
ID: A
9. ANS: A
2x 2 − 48
x+6
−
(x − 4) (x + 4) x + 4
=
2x 2 − 48
x+6
−
(x − 4) (x + 4) x + 4
Factor the denominators.
ÊÁ x − 4 ˆ˜
ÁÁ
˜
ÁÁ x − 4 ˜˜˜
Ë
¯
2x 2 − 48 − (x + 6) (x − 4)
(x − 4) (x + 4)
Ê
ˆ
2x 2 − 48 − ÁÁ x 2 + 2x − 24 ˜˜
Ë
¯
=
(x − 4) (x + 4)
=
=
2x 2 − 48 − x 2 − 2x + 24
(x − 4) (x + 4)
x 2 − 2x − 24
(x − 4) (x + 4)
(x − 6) (x + 4) x − 6
=
=
(x − 4) (x + 4) x − 4
=
The LCD is (x − 4) (x + 4) , so multiply
x+6
x−4
by
.
x+4
x−4
Subtract the numerators.
Multiply the binomials in the numerator.
Distribute the negative sign.
Write the numerator in standard form.
Factor the numerator, and divide out common factors.
The expression is undefined at x = 4 and x = −4 because these values of x make the factors (x − 4) and (x + 4)
equal 0.
Feedback
A
B
C
D
Correct!
Check your distribution of the negative sign.
Did you factor the numerator and divide out common factors correctly?
Did you factor the numerator and divide out common factors correctly?
PTS:
OBJ:
STA:
10. ANS:
OBJ:
TOP:
11. ANS:
REF:
OBJ:
NAT:
KEY:
12. ANS:
REF:
OBJ:
NAT:
KEY:
1
DIF: Average
REF: Page 585
8-3.4 Subtracting Rational Expressions
NAT: 12.5.3.c
2A.2.A
TOP: 8-3 Adding and Subtracting Rational Expressions
A
PTS: 1
DIF: L2
REF: 12-1 Graphing Rational Functions
12-1.1 Graphing Rational Functions
NAT: ADP J.1.6 | ADP J.2.2 | ADP J.2.3
12-1 Example 1
KEY: rational function | constant of variation | inverse variation
D
PTS: 1
DIF: L2
12-2 Simplifying Rational Functions
12-2.1 Simplifying Rational Expressions
NAEP 2005 A3c | ADP J.1.5 | ADP J.1.6
TOP: 12-2 Example 2
rational expression
A
PTS: 1
DIF: L2
12-2 Simplifying Rational Functions
12-2.1 Simplifying Rational Expressions
NAEP 2005 A3c | ADP J.1.5 | ADP J.1.6
TOP: 12-2 Example 2
rational expression
6
ID: A
13. ANS:
REF:
OBJ:
NAT:
KEY:
14. ANS:
REF:
OBJ:
NAT:
KEY:
15. ANS:
REF:
OBJ:
NAT:
KEY:
16. ANS:
REF:
OBJ:
NAT:
KEY:
17. ANS:
REF:
OBJ:
NAT:
KEY:
18. ANS:
REF:
OBJ:
NAT:
TOP:
19. ANS:
REF:
OBJ:
NAT:
TOP:
20. ANS:
REF:
OBJ:
NAT:
TOP:
C
PTS: 1
DIF: L3
12-3 Multiplying and Dividing Rational Expressions
12-3.1 Multiplying Rational Expressions
NAEP 2005 A3b | NAEP 2005 A3c | ADP J.1.5
TOP: 12-3 Example 1
rational expression
B
PTS: 1
DIF: L2
12-3 Multiplying and Dividing Rational Expressions
12-3.1 Multiplying Rational Expressions
NAEP 2005 A3b | NAEP 2005 A3c | ADP J.1.5
TOP: 12-3 Example 2
rational expression
D
PTS: 1
DIF: L3
12-3 Multiplying and Dividing Rational Expressions
12-3.1 Multiplying Rational Expressions
NAEP 2005 A3b | NAEP 2005 A3c | ADP J.1.5
TOP: 12-3 Example 2
rational expression
A
PTS: 1
DIF: L2
12-3 Multiplying and Dividing Rational Expressions
12-3.2 Dividing Rational Expressions
NAEP 2005 A3b | NAEP 2005 A3c | ADP J.1.5
TOP: 12-3 Example 4
rational expression
C
PTS: 1
DIF: L2
12-3 Multiplying and Dividing Rational Expressions
12-3.2 Dividing Rational Expressions
NAEP 2005 A3b | NAEP 2005 A3c | ADP J.1.5
TOP: 12-3 Example 4
rational expression
D
PTS: 1
DIF: L2
12-5 Adding and Subtracting Rational Expressions
12-5.1 Adding and Subtracting Rational Expressions With Like Denominators
NAEP 2005 N5b | NAEP 2005 A3b | NAEP 2005 A3c | ADP J.1.5
12-5 Example 1
KEY: rational expression
D
PTS: 1
DIF: L3
12-5 Adding and Subtracting Rational Expressions
12-5.1 Adding and Subtracting Rational Expressions With Like Denominators
NAEP 2005 N5b | NAEP 2005 A3b | NAEP 2005 A3c | ADP J.1.5
12-5 Example 2
KEY: rational expression
B
PTS: 1
DIF: L2
12-5 Adding and Subtracting Rational Expressions
12-5.2 Adding and Subtracting Rational Expressions With Unlike Denominators
NAEP 2005 N5b | NAEP 2005 A3b | NAEP 2005 A3c | ADP J.1.5
12-5 Example 4
KEY: rational expression
7
ID: A
21. ANS: C
a 2 + 14a + 48
(a + ?) (a + ?)
(a + 6) (a + 8)
Look for the factors of 48 whose sum is 14.
The factors are 6 and 8.
Feedback
A
B
C
D
Look for factors whose product is the trinomial's last term.
Use the FOIL method to check your answer.
Correct!
Use the FOIL method to check your answer.
PTS: 1
DIF: Basic
REF: Page 541
OBJ: 8-3.2 Factoring x^2 + bx + c When c is Positive
NAT: 12.5.3.d
TOP: 8-3 Factoring x^2 + bx + c
22. ANS: D
Since a = 2, the coefficients of the First terms must be factors of 2.
Since c = 6, the Last terms must be factors of 6.
Since b = 7, the Outer and Inner products must add up to 7.
The sum of the products of the outer and inner terms should be 7.
It may be helpful to make a table to check all the factors of 2 and all the factors of 6. Then check the products
of the outer and inner terms to see if the sum is 7.
Feedback
A
B
C
D
You reversed the second terms in the parentheses.
When b is negative, the factors of c are both negative. When b is positive, the factors of
c are both positive.
The coefficient of the x-term in the second binomial cannot be 1. Check your answer.
Correct!
PTS: 1
DIF: Basic
REF: Page 549
OBJ: 8-4.2 Factoring ax^2 + bx + c When c is Positive
NAT: 12.5.3.d
TOP: 8-4 Factoring ax^2 + bx + c
23. ANS: D
x1 ⋅ y1 = x2 ⋅ y2
Write the Product Rule for Inverse Variation.
15 ⋅ 8 = x 2 ⋅ 5
Substitute 15 for x 1 , 8 for y 1 , and 5 for y 2 .
x 2 = 24
Simplify and solve for x 2 .
Feedback
A
B
C
D
Divide both sides of the equation by the same number, not subtract.
The Product Rule for Inverse Variation states that (x1)(y1) = (x2)(y2).
The Product Rule for Inverse Variation states that (x1)(y1) = (x2)(y2).
Correct!
PTS: 1
NAT: 12.5.1.e
DIF: Basic
STA: A.11.B
REF: Page 853
OBJ: 12-1.4 Using the Product Rule
TOP: 12-1 Inverse Variation
8
ID: A
24. ANS: B
x−3
x−3
=
2
(x
−
3)(x − 2)
x − 5x + 6
x−3
=
(x − 3)(x − 2)
1
=
x−2
Factor the numerator and denominator.
Divide out the common factors.
Simplify.
Feedback
A
B
C
D
Factor the denominator. Divide out common factors.
Correct!
Factor the denominator. Divide out common factors.
Factor the denominator. Divide out common factors.
PTS: 1
DIF: Basic
REF: Page 867
OBJ: 12-3.3 Simplifying Rational Expressions with Trinomials
NAT: 12.5.3.c
STA: A.4.A
TOP: 12-3 Simplifying Rational Expressions
25. ANS: A
x2 − x − 6
x2 + x
⋅
2x 2 − 6x x 2 + 4x + 4
(x + 2)(x − 3)
x(x + 1)
⋅
=
Factor the numerator and denominator.
2x(x − 3)
(x + 2)(x + 2)
1 (x + 1)
= ⋅
Simplify.
2 (x + 2)
x+1
=
Multiply the remaining factors.
2x + 4
Feedback
A
B
C
D
Correct!
Factor the numerator and denominator and divide out the common factors.
Factor the numerator and denominator and divide out the common factors.
Factor the numerator and denominator and divide out the common factors.
PTS: 1
DIF: Average
REF: Page 879
OBJ: 12-4.3 Multiplying Rational Expressions Containing Polynomials
NAT: 12.5.3.c
STA: A.4.B
TOP: 12-4 Multiplying and Dividing Rational Expressions
9
ID: A
26. ANS: A
1 m− 8
÷
m
8m
1 8m
= ⋅
m m− 8
1(8m)
=
m(m − 8)
8
=
m−8
Write as multiplication by the reciprocal.
Multiply the numerators and the denominators.
Divide out common factors. Simplify.
Feedback
A
B
C
D
Correct!
Divide out common factors, and simplify.
First, write as multiplication by the reciprocal. Then, multiply the numerators and the
denominators.
Write as multiplication by the reciprocal first.
PTS: 1
DIF: Basic
REF: Page 880
OBJ: 12-4.4 Dividing by Rational Expressions and Polynomials
NAT: 12.5.3.c
STA: A.4.A
TOP: 12-4 Multiplying and Dividing Rational Expressions
27. ANS: A
m2 + 10m + 24
m+4
Factor the numerator.
(m + 4) (m + 6)
=
m+4
= m+ 6
Divide out the common factors. Simplify.
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Correct!
First, factor the numerator. Then, divide out the common factors and simplify.
Check the signs.
First, factor the numerator. Then, divide out the common factors and simplify.
PTS: 1
DIF: Basic
REF: Page 894
OBJ: 12-6.2 Divide a Polynomial by a Binomial
STA: A.4.A
TOP: 12-6 Dividing Polynomials
10
NAT: 12.5.3.c