Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
1)
Tom’s teacher asks him to solve the following problem: “The product of two consecutive
even numbers is 168. Find the numbers.” Tom rewords the problem and translates to the
following equation:
x(x + 1) = 168
Why is this equation not correct and what would the correct equation be?
1)
2)
In factoring a trinomial in y as (y+ a)(y+ b), what must be true of a and b, if the coefficient
of the last term of the trinomial is negative?
2)
3)
The height of an object after t seconds is given by the equation h = – 16t2+ 10t + 7. When
h = 0, solving for t means finding the time when the object hits the ground. A student uses
the quadratic formula and finds 2 solutions. One is negative and the other positive. Which
solution makes sense? Which does not? Why?
3)
4)
A student is solving the equation s2=92 and has determined that s =9. He or she has
looked up the answer in the back of the book and finds that s = – 9 is also a solution. The
student feels that this is a misprint. How would you advise him or her?
4)
5)
Suppose you have to solve the following problem: “The length of a rectangle is 5 ft more
than its width and its area is 84 square feet. Find its width and length.” Let x represent the
width of the rectangle and translate the problem into an equation. Do not solve the
equation.
5)
6)
Maria’s teacher asked her to solve the following problem: “One leg of a right triangle is 5 m
longer than the other. The length of the hypotenuse is 25 m. Find the lengths of the legs.”
Maria called the legs x and y and translated the problem into the following equation:
x2+y2=252. After that she was unsure of how to proceed. Why is Maria’s equation not
very useful? Write the correct equation and find the solution to the problem showing all
the steps of your work.
6)
7)
Why is 5 called a triple solution to the equation (x – 5)3= 0?
7)
8)
Jason is given the following information: “The shortest side of a triangle is 4 cm less than
the middle side. The length of the longest side is 13 cm.” Jason claims that he can find the
lengths of the two shorter sides by solving the following equation:
x2+(x + 4)2=132
What is wrong with his reasoning?
8)
9)
Brenda’s teacher asks her to solve the following problem: “The product of two consecutive
numbers is 210. Find the numbers.” Brenda knows that she must translate this problem into
an equation. What equation could Brenda write to represent this problem?
9)
10)
What steps would you take to factor x2– 5x + 6 ?
10)
2
11)
How could you solve the equation
(10x + 10)(9x – 2)(10x – 4) = 0? How many equations do you need to solve? What are their
solutions?
11)
12)
Tina was asked to solve the following problem: “The product of two consecutive numbers
is 90. Find the numbers.” Tina’s solution is given below. Do you agree with her conclusion?
If not, why not?
x(x + 1) = 90
x2+ x = 90
x2+ x – 90 = 0
(x + 10)(x – 9) = 0
x = – 10 or x = 9
The pair of numbers is 9 and 10
12)
13)
Explain the error in the following:
x2+ 2x – 15 = (x – 5)(x + 3)
13)
x2+ 2x – 15 = (x + 5)(x – 3)
14)
Give an example of three numbers whose greatest common factor is 9.
14)
15)
A student is trying to solve the equation (x + 4)(x – 5) =10. The student has set x + 4 =10
and x – 5 =10 and found that two solutions x =6, x =15. The student checks his or her
results by plugging in his or her solutions into the original equation and finds that they do
not work. How would you advise him or her?
15)
16)
A student is told that there are two solutions to the problem 7x2=7x. The student can
only find one; that is, x = 1. How could you advise this student?
16)
17)
A student was trying to solve the problem 4x(7x – 3) = 0. The student knew that he or she
should set 7x – 3 = 0 but was confused about whether or not he or she should set 4x = 0, or
4= 0 and x = 0. How would you advise this student?
17)
18)
A student is solving the equation x2=2x This student has decided to divide both sides of
the equation by x and finds the solution x =2. The student checks the answer in the back of
the book and finds that x = 0 is also a solution. The student feels that this is a misprint.
How would you advise him or her?
18)
Explanation:
19)
For the polynomial 18x2+ 19x + 18, 2 is not a common factor. Explain why the binomial
2x – 9, then, cannot be a factor of the polynomial.
19)
Explanation:
Solve the problem.
20)
On a quiz, a student factored 8y3+72y2+144y by first factoring out 8y to get
8y(y2+9y +18). He then wrote the answer (y +6)(y +3) as the factored answer. Why is this
answer wrong?
20)
Explanation:
Provide an appropriate response.
21)
If an object is dropped, the distance it falls after t seconds is given by d =1
2gt2. A student is
determining how long it would take an object to fall 46 feet on planet x having gravity 16
ft/sec2. The student determines the two solutions t = ± 23
4. Are both correct answers? Why
or why not?
21)
Explanation:
Answer the question.
22)
Why is the answer (x2– 81)(x2+ 81) not the correct answer to the instruction “Factor
(x4– 6561) completely“?
22)
Explanation:
4
Explanation:
Provide an appropriate response.
23)
Write a problem in which the quadratic equation x(x + 3) = 108 must be solved in order to
solve the problem.
23)
24)
What steps would you take to factor x2+ 12x + 35 ?
24)
25)
Use the FOIL method to show that (4x + 8)(x – 7) is 4x2– 20x – 56. If you were asked to
completely factor 4x2– 20x – 56, why would it be incorrect to give (4x + 8)(x – 7) as your
answer?
25)
26)
Mark’s teacher asked him to solve the following problem: “One leg of a right triangle is 7
m longer than the other. The length of the hypotenuse is 13 m. Find the lengths of the legs.”
Mark translated the problem into the following equation: (x2+ 7) +x2=132. Why is this
equation not correct and what would the correct equation be?
26)
Answer the question.
27)
The binomial 25x2+ 100 is the sum of two squares that can be factored. Why is this
possible?
27)
Provide an appropriate response.
28)
A student is told that there are two solutions to the problem 5x2=8x. The student can
only find one, namely, x =8
5. How can you advise him or her?
28)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Factor.
29)
29)
A)
(4m – 5)(16m2+ 20m + 25)
B)
(4m + 5)(16m2+ 25)
C)
(64m + 5)(m2– 20m + 25)
D)
(4m + 5)(16m2– 20m + 25)
5
Solve the equation.
30)
30)
A)
{9, 4}
B)
{–9, 1}
C)
{–9, 4}
D)
{9, –4}
31)
31)
A)
5
6, –5
6
B)
–5
6, 0
C)
5
6, 0
D)
{0}
C
Decide whether or not the correct factored form of the polynomial is given.
32)
32)
A)
No
B)
Yes
A
Factor as completely as possible. If unfactorable, indicate that the polynomial is prime.
33)
33)
A)
(12x – 3t)(x – 2t)
B)
(4x – 3t)(3x – 2t)
C)
(4x + 3t)(3x + 2t)
D)
prime
B
34)
34)
A)
(9z + 2)(z – 4)
B)
(3z + 2)(3z – 4)
C)
prime
D)
(3z – 2)(3z + 4)
B
C
Factor completely.
35)
35)
A)
x5(x –2y)(x +5y)
B)
x5(x –5y)(x +2y)
C)
x4(x –5y)(x +2y)
D)
x5(x +5y)(x +2y)
Factor by grouping.
36)
36)
A)
(5 +s)(7–t)
B)
(5 +s)(7+t)
C)
(5 –s)(7–t)
D)
(5 –s)(7+t)
D)
Factor the polynomial.
37)
37)
A)
(k2+16)(k2–2)
B)
(k +4)(k –4)(k2–2)
C)
(k –4)2(k2+2)
D)
(k –4)2(k2–2)
D)
Factor.
38)
38)
A)
4(9x + 5y)2
B)
–4(9x + 5y)(9x – 5y)
C)
–4(9x – 5y)2
D)
–4(9x + 5y)2
D)
Factor as completely as possible. If unfactorable, indicate that the polynomial is prime.
39)
39)
A)
(27x + 18)(x – 5)
B)
9(3x + 2)(x – 5)
C)
prime
D)
9(3x – 2)(x + 5)
D)
D)
Factor.
40)
40)
A)
(x + 10)2
B)
Prime
C)
(x + 10)(x – 10)
D)
(x – 10)2
Factor as completely as possible. If unfactorable, indicate that the polynomial is prime.
41)
41)
A)
y2(x – 1)(20x + 3)
B)
y2(5x – 1)(4x + 3)
C)
(4x – 1y)(5x + 3y)
D)
(5x – 1y)(4x + 3y)
B
Solve the problem.
42)
42)
A)
12 cm
B)
4 cm
C)
2 cm
D)
6 cm
D
Provide an appropriate response.
43)
43)
A)
Yes
B)
No
B
Solve the equation.
44)
44)
A)
2
5, 2
3
B)
–5
2, –3
2
C)
5
2, 3
2
D)
–2
5, –3
2
B
8
A
Factor.
45)
45)
A)
(m – 343)(m2+ 1)
B)
(m + 7)(m2– 7m + 49)
C)
(m – 7)(m2+ 7m + 49)
D)
(m – 7)(m2+ 49)
Factor out the greatest common factor, or a negative common factor if the coefficient of the term of greatest degree is
negative.
46)
46)
A)
–7s3(s3+ 4)
B)
–s3(7s3+28)
C)
21(s3+ 4s)
D)
–7s3(s3– 4)
Factor completely.
47)
47)
A)
Prime
B)
x(x + 7)(x – 6)
C)
(x2+ 1)(x – 42)
D)
x(x + 6)(x – 7)
Factor out the greatest common factor.
48)
48)
A)
t(3– m) + s
B)
no common factor (except 1)
C)
(t – s)(3– m)
D)
(t + s)(3– m)
Factor as completely as possible. If unfactorable, indicate that the polynomial is prime.
49)
49)
A)
(–5x + 1)(x + 2)
B)
–(5x + 1)(x + 2)
C)
–(5x – 1)(x – 2)
D)
Prime
Factor the polynomial.
50)
50)
A)
(7q – p)(8q – p)
B)
(56q – p)(q – p)
C)
(7q + p)(8q – p)
D)
(pq –7)(pq –8)
Solve the problem.
51)
51)
A)
71,628,000
B)
68,852,000
C)
1,490,000
D)
1,490,000,000,000
Factor completely.
52)
52)
A)
x3(x – 1)(5x + 20)
B)
53(x2+ 3x – 4)
C)
x3(5x – 5)(x + 4)
D)
5x3(x – 1)(x + 4)
Solve the problem.
53)
53)
A)
0, 1 or 17, 18
B)
0, 1 or 18, 19
C)
0, 1
D)
17, 18
Answer the question.
54)
54)
A)
It is not possible for any n.
B)
n must be a multiple of 2.
C)
n must be a multiple of 5.
D)
n must be a multiple of 3.
Factor completely.
55)
55)
A)
Prime
B)
(x – 10)(x + 3)
C)
(x + 30)(x – 1)
D)
(x + 10)(x – 3)
A
Solve the equation.
56)
56)
A)
–3
2, 5
4, 0
B)
– 1, 1
C)
–3
4, 5
2
D)
{0}
A
Factor the polynomial.
57)
57)
A)
(3x + y + 5)2
B)
prime
C)
(3x + y + 5)(3x + y – 5)
D)
(3x + y – 5)2
C
Complete the factoring.
58)
58)
A)
x – 3
B)
7x + 3
C)
3x + 7
D)
x + 7
B
D
Factor the polynomial.
59)
59)
A)
4x2(9x –5)
B)
4x(9x –5)
C)
x(36x –20)
D)
4x(9x –20)
Solve the problem.
60)
60)
A)
48 units
B)
288 units
C)
12 units
D)
72 units
Provide an appropriate response.
61)
61)
A)
No
B)
Yes: 20x5(y + 11)
C)
Yes: 20x5(y + 4) + 7
D)
Yes: (y + 4)(20x5+ 7)
Solve the problem.
62)
62)
A)
108 yd
B)
7.4 yd
C)
18 yd
D)
14.7 yd
Factor as completely as possible. If unfactorable, indicate that the polynomial is prime.
63)
63)
A)
prime
B)
(10x – 5)(x + 4)
C)
5(2x + 1)(x – 4)
D)
5(2x – 1)(x + 4)
Find the greatest common factor of the terms.
64)
64)
A)
448a9b9
B)
8a9b9
C)
8a6b2
D)
4a3b7
C
Find the greatest common factor of the numbers.
65)
65)
A)
3
B)
2
C)
1
D)
6
A
Factor.
66)
66)
A)
(7x – 5y)(7x + 5y)
B)
(7x – 5y)2
C)
Prime
D)
(7x + 5y)2
B
Factor completely.
67)
67)
A)
Prime
B)
5(x + y)(x – 4y)
C)
(5x – 5y)(x + 4y)
D)
5(x – y)(x + 4y)
B
C
Factor the polynomial.
68)
68)
A)
2t(5t2+ t –8)
B)
2t(5t2+2t –16)
C)
t(10t2+2t –16)
D)
2(5t3+t2–8t)
Factor as completely as possible. If unfactorable, indicate that the polynomial is prime.
69)
69)
A)
(3z + 2)(5z – 2)
B)
prime
C)
(15z + 2)(z – 2)
D)
(3z – 2)(5z + 2)
Solve the equation.
70)
70)
A)
–2
3, –3
5
B)
3
2, 5
3
C)
2
3, 3
5
D)
–3
2, –3
5
Factor as completely as possible. If unfactorable, indicate that the polynomial is prime.
71)
71)
A)
prime
B)
(3x – t)(4x – 3t)
C)
(3x + t)(4x + 3t)
D)
(12x + t)(x + 3t)
The middle term of the trinomial has been rewritten. Now factor by grouping.
72)
72)
A)
(2x – 5)(3x – 4)
B)
(6x + 5)(x + 4)
C)
(2x + 5)(3x + 4)
D)
(6x – 5)(x – 4)
14
Complete the factoring.
73)
73)
A)
x + 8
B)
x2+ 8
C)
x + 42
D)
x – 20
Solve the problem.
74)
74)
A)
2.7 mi
B)
14.5 mi
C)
3.5 mi
D)
5.4 mi
D
Solve the equation.
75)
75)
A)
–1, 0
B)
{0}
C)
–1
D)
–1, 84
A
Factor the polynomial.
76)
76)
A)
(n +m)(4– t)
B)
(n –4)(m+ t)
C)
(n – t)(m+4)
D)
(n +4)(m– t)
D
15
A
Solve the equation.
77)
77)
A)
3, –7
4, –2
9
B)
3, 7
4, 2
9
C)
{3}
D)
–9
4, –2
9
Factor out the greatest common factor, or a negative common factor if the coefficient of the term of greatest degree is
negative.
78)
78)
A)
–5x(x +30)
B)
–5x2(x +6)
C)
–5x(x –6)
D)
x(5x +30)
Factor the polynomial.
79)
79)
A)
(6m – 8)(9m2+ 16)
B)
2(3m + 4)(9m2– 12m + 16m)
C)
2(3m – 4)(9m2+ 12m + 16)
D)
2(27m – 4)(m2+ 12m + 16)
Find the greatest common factor of the numbers.
80)
80)
A)
1
B)
11
C)
5
D)
3
Factor.
81)
81)
A)
(2p – 1)(4p2+ 1)
B)
(2p – 1)(4p2+ 2p + 1)
C)
(8p – 1)(p2+ 2p + 1)
D)
(2p + 1)(4p2– 2p + 1)