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MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1 - 10.1, but not 8.1 or 8.2)
Test also includes review problems from earlier sections so study test reviews 1 and 2 also.
NOTE: In addition to the problems below, please study the handout Exercise Set 10.1 and
review earlier sections.
1.
Simplify: 57 • 52
2.
Simplify: (–6ab5c3)(–3a2c5)
3.
Simplify: –4x2yz3 • 7wxyz2
4.
Simplify: y16 ÷ y9
5.
Simplify:
310
36
6.
Simplify:
14w 2x 4 y 6z0
21w 8 x 4 y3z5
7.
Simplify: (114)6
8.
Simplify: (–4xy4)3
9.
Simplify: (–2a2b5)4
10.
Simplify: (5v5w3)(–2v2w5)3
11.
Simplify: 40
12.
Simplify: 5x0
13.
Simplify: (5x)0
14.
Identify each polynomial below as a monomial, a binomial, or a trinomial and
indicate its degree.
a.
10 + 3z4 – 9z3
b.
19m15
c.
x + 2x5 – 4
d.
100 – 5y
e.
–8
15.
Evaluate 2u3v – 5uv + 3u – v + 10 when u = 2 and v = –5.
16.
Add and simplify: (11x3 + 4x2 – 5y3 – 3y) + (2x3 – 4x2 + y)
17.
Subtract and simplify: (7y2 – 2x + y) – (4y2 – 3y + 2x)
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1 - 10.1, but not 8.2)
18.
Multiply and simplify: 3xy3(5x3 – 7xy2 + xy – 2y)
19.
Divide:
12xy 6 + 24 x5 y9
4xy3
20.
Divide:
20r3s2t – 4rs2t
12r2st3
21.
Divide: (12x2 – 7x – 13) ÷ (3x + 2)
22.
Divide: (x3 – 11x + 6) ÷ (x – 3)
23.
Divide: (3x3 + 4x2 + 8) ÷ (x + 2)
24.
Multiply and simplify: (w – 5)(w – 7)
25.
Multiply and simplify: (8y + 1)(2y – 7)
26.
Multiply and simplify: (2y + 3)2
27.
Multiply and simplify: (w – 4)2
28.
Multiply and simplify: (6r – 7s)2
29.
Multiply and simplify: (3x – 4)(3x + 4)
30.
Multiply and simplify: (w + 6v)(w – 6v)
31.
Multiply and simplify: (5x – 3)(2y + 9)
32.
Multiply and simplify: (4a3 – 3)(a3 – 5a + 1)
33.
Factor completely: a2 – b2
34.
Factor completely: a2 + b2
35.
Factor completely: a2 – 2ab + b2
36.
Factor completely: a2 + 2ab + b2
37.
Factor completely
12 x 2 − 40 xy + 25 y 2
38.
Factor completely:
15x 2 − 13x − 72
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1 - 10.1, but not 8.2)
39.
Factor completely: 9ab3 – 6a2b2
40.
Factor completely: 10a3 + 6a2b – 2a
41.
Factor completely: 21x3y2z4 – 28x5y3
42.
Factor completely: 12x2 – 3xy – 8x + 2y
43.
Factor completely: 5x2 + 5xy – x – y
44.
Factor completely: x2 – 9x + 20
45.
Factor completely: x2 – 15x – 14
46.
Factor completely: 2y2 + 4y – 48
47.
Factor completely: 2x2 + 7x + 6
48.
Factor completely: 6y2 + 7y – 20
49.
Factor completely: 2x3 – 7x2 – 30x
50.
Factor completely: 3x2 – 16x + 5
51.
Factor completely: 9a2 + 42a + 49
52.
Factor completely: y2 – 12y + 36
53.
Factor completely: 25x2 – 10x + 1
54.
Factor completely: 2x3 – 18x
55.
Factor completely: 49a2 – 81b2
56.
Factor completely: 25x2 + 36y2
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1 - 10.1, but not 8.2)
NOTE: The equations in Problems 60 – 70 are quadratic equations, which means they can be
written in the form ax2 + bx + c = 0 where a ≠ 0. To solve them, simplify each side of the
equation if possible, then move all terms to one side of the equation so that one side equals
zero, and then solve using one of the following two methods: factoring or quadratic formula
(when all else fails use this latter method). You will learn about other methods in later math
courses. If you have trouble, please ask your instructor.
57.
Solve for y: 2y2 – 8y – 24 = 0
58.
Solve for x: x2 + 2x = 15
59.
Solve for a: a2 – 4a + 10 = 3a
60.
Solve for m: 1 – 3m2 = m2 – 2(m + 1)
61.
Solve for n: 3n2 + 1 = 9 – 2n
62.
Twice a number is fifteen less than the square of that number. Find the number.
63.
The product of two consecutive integers is 210. Find the two integers.
64.
The sum of the squares of two consecutive odd integers is three less than eleven times
the larger. Find the two integers.
65.
A triangle has a base that is 10 cm more than its height. The area of the triangle is 12
square cm. Find the height and the base.
66.
Find the dimensions of a rectangular picture whose length is 3 inches shorter than twice
its width and whose area is 35 square inches.
67.
Find the lengths of the sides of a right triangle if the long leg is 7 cm longer than the
short leg and the hypotenuse is 1 cm longer than the long leg.
68. A 113–meter rope is divided into two pieces so that one piece is fifteen meters longer
than the other piece. Find the length of each piece.
69. Review solving equations with denominators, graphing, slope of a line, word problems
that can be solved by a system of equations, word problems involving area and
perimeter, and solving formulas for one of the variables.
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1 - 10.1, but not 8.2)
ANSWERS:
1.
59
2.
18a3b5c8
3.
–28wx3y2z5
4.
y7
5.
34
6.
2y3
3w 6z5
7.
1124
8.
–64x3y12
9.
16a8b20
10.
–40v11w18
11.
1
12.
5
13.
1
14.
a.
b.
c.
d.
e.
15.
–9
16.
13x3 – 5y3 – 2y
17.
3y2 – 4x + 4y
18.
15x4y3 – 21x2y5 + 3x2y4 – 6xy4
19.
3y3 + 6x4y6
trinomial, 4
monomial, 15
trinomial, 5
binomial, 1
monomial, 0
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1 - 10.1, but not 8.2)
ANSWERS:
20.
21.
5rs
s
–
3t 2
3rt 2
-3
3
4x – 5 +
, or 4x – 5 –
3x + 2
3x + 2
22.
x2 + 3x – 2
23.
3x2 – 2x + 4
24.
w2 – 12w + 35
25.
16y2 – 54y – 7
26.
4y2 + 12y + 9
27.
w2 – 8w + 16
28.
36r2 – 84rs + 49s2
29.
9x2 – 16
30.
w2 – 36v2
31.
10xy + 45x – 6y – 27
32.
4a6 – 20a4 + a3 + 15a – 3
33.
(a – b)(a + b) [NOTE: Know this formula, and use it for factoring a difference of two
squares.]
34.
Prime (Not Factorable) [NOTE: Know this formula. A sum of two squares is not
factorable unless it has a common factor. If it has a common factor, factor it out.]
35.
(a – b)2 [NOTE: Know this formula, and use it for factoring a perfect square trinomial
with a negative middle term.]
36.
(a + b)2 [NOTE: Know this formula, and use it for factoring a perfect square trinomial
with a positive middle term.]
37.
(2x-5y)(6x-5y)
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1 - 10.1, but not 8.2)
ANSWERS:
38.
(3x-8)(5x+9)
39.
3ab2(3b – 2a)
40.
2a(5a2 + 3ab – 1)
41.
7x3y2(3z4 – 4x2y)
42.
(4x – y)(3x – 2)
43.
(x + y)(5x – 1)
44.
(x – 4)(x – 5)
45.
Prime (Not Factorable) [NOTE: (x – 14)(x – 1) = x2 – 15x + 14, not x2 – 15x – 14]
46.
2(y + 6)(y – 4)
47.
(2x + 3)(x + 2)
48.
(2y + 5)(3y – 4)
49.
x(2x + 5)(x – 6)
50.
(3x – 1)(x – 5)
51.
(3a + 7)2
52.
(y – 6)2
53.
(5x – 1)2
54.
2x(x + 3)(x – 3)
55.
(7a + 9b)(7a – 9b)
56.
Prime (Not Factorable) - Sum of Squares [NOTE: If you got a different answer,
carefully multiply it back together to see that it is not equal to the original problem.]
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1 - 10.1, but not 8.2)
ANSWERS:
57.
y = 6, or y = –2
58.
x = 3, or x = –5
59.
a = 5, or a = 2
60.
m=
1 + 13
1 − 13
≈ 1.151, or m =
≈ −0.651
4
4
61.
n=
4
, or n = −2
3
62.
The number is 5 or –3. (NOTE: You must give both parts of the answer.)
To set up this problem, let x = the number. Then the equation is 2x = x2 – 15.
63.
The integers are 14 and 15, or the integers are –15 and –14. (NOTE: You must give
both parts of the answer.)
To set up this problem, let x = the first integer. Then x + 1 = the next consecutive
integer. The equation is x(x + 1) = 210.
64.
The integers are 5 and 7. (NOTE: –1.5 and 0.5 are not integers.)
To set up this problem, let x = the first integer. Then x + 2 = the next consecutive odd
integer. The equation is x2 + (x + 2)2 = 11(x + 2) – 3.
65.
The height of the triangle is 2 cm and the base is 12 cm.
To set up this problem, let x = the base. Then x – 10 = the height. The formula for area
of a triangle is A =
1
2
bh so the equation is
1
2
x(x – 10) = 12. Hint: Multiply both sides of
the equation by 2 to clear the fraction, which makes the equation x(x – 10) = 24.
66.
The width of the rectangle is 5 inches and the length is 7 inches.
To set up this problem, let x = the width. Then 2x – 3 = the length. The formula for area
of a rectangle is A = lw so the equation is (2x – 3)x = 35, or x(2x – 3) = 35.
67.
The lengths of the sides of the triangle are 5 cm, 12 cm, and 13 cm. (NOTE: The
hypotenuse of a right triangle is always the longest side.)
To set up this problem, let x = the length of the short leg. Then x + 7 = the length of the
long leg, and x + 7 + 1 = x + 8 = the length of the hypotenuse. Use the Pythagorean
Theorem a2 + b2 = c2 to write the equation x2 + (x + 7)2 = (x + 8)2.
68.
The lengths of the pieces of rope are 49 m and 64 m.
To set up this problem, let x = the length of the short piece. Then x + 15 = the length of
the long piece. The sum of the two pieces is the total length of rope, which means that
the equation is x + (x + 15) = 113.