Download NOTE: In addition to the problems below, please study the handout

MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 3 (New Material From: 6.1-6.3, 7.1-7.3, and 10.1)
NOTE: In addition to the problems below, please study the handout Exercise Set 10.1 posted
at http://www.austincc.edu/jbickham/handouts.
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 y3 z5
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 3 (New Material From: 6.1-6.3, 7.1-7.3, and 10.1)
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: 9ab3 – 6a2b2
38.
Factor completely: 10a3 + 6a2b – 2a
39.
Factor completely: 21x3y2z4 – 28x5y3
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 3 (New Material From: 6.1-6.3, 7.1-7.3, and 10.1)
40.
Factor completely: 12x2 – 3xy – 8x + 2y
41.
Factor completely: 5x2 + 5xy – x – y
42.
Factor completely: x2 – 9x + 20
43.
Factor completely: x2 – 15x – 14
44.
Factor completely: 2y2 + 4y – 48
45.
Factor completely: 2x2 + 7x + 6
46.
Factor completely: 6y2 + 7y – 20
47.
Factor completely: 2x3 – 7x2 – 30x
48.
Factor completely: 3x2 – 16x + 5
49.
Factor completely: 9a2 + 42a + 49
50.
Factor completely: y2 – 12y + 36
51.
Factor completely: 25x2 – 10x + 1
52.
Factor completely: 2x3 – 18x
53.
Factor completely: 49a2 – 81b2
54.
Factor completely: 25x2 + 36y2
NOTE: The equations in Problems 55 – 69 are quadratic equations, which means they can be
written in the form ax2 + bx + c = 0 where a ≠ 0. Solve these equations using one of the
following three methods: by using the square root property, by factoring, or by using the
quadratic formula (when all else fails use this latter method - it should always work on any of
these problems). The quadratic formula will be provided on your test. To solve using the
square root property, refer to lesson 10.1. To solve by factoring or by using the quadratic
formula, simplify each side of the equation if possible, then move all terms to one side of the
equation so that one side of the equation equals zero, and then factor the other side of the
equation or use the quadratic formula. Please refer to lesson 10.1 and the handout Exercise
Set 10.1 posted at http://www.austincc.edu/jbickham/handouts for more information. If you
have trouble, please ask for help.
55.
Solve for b: b2 = 81
56.
Solve for x: 2x2 = 98
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 3 (New Material From: 6.1-6.3, 7.1-7.3, and 10.1)
57.
Solve for a: 5a2 – 180 = 0
58.
Solve for w: (w – 3)2 = 25
59.
Solve for y: 2y2 – 8y – 24 = 0
60.
Solve for x: x2 + 2x = 15
61.
Solve for a: a2 – 4a + 10 = 3a
62.
Solve for m: 1 – 3m2 = m2 – 2(m + 1)
63.
Solve for n: 3n2 + 1 = 9 – 2n
64.
Twice a number is fifteen less than the square of that number. Find the number.
65.
The product of two consecutive integers is 210. Find the two integers.
66.
The sum of the squares of two consecutive odd integers is three less than eleven times
the larger. Find the two integers.
67.
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.
68.
Find the dimensions of a rectangular picture whose length is 3 inches shorter than twice
its width and whose area is 35 square inches.
69.
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.
70.
The length of one leg of a right triangle is 14 feet. The length of the hypotenuse is 7
feet longer than the other leg. Find the lengths of the hypotenuse and the other leg.
71.
A 114–meter rope is divided into two pieces so that one piece is three times as long as
the other piece. Find the length of each piece.
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 3 (New Material From: 6.1-6.3, 7.1-7.3, and 10.1)
ANSWERS:
1.
59
2.
18a3b5c8
3.
–28wx3y2z5
4.
y7
5.
34
6.
2y3
3w 6 z5
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 3 (New Material From: 6.1-6.3, 7.1-7.3, and 10.1)
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.
a2 – b2 = (a – b)(a + b) [NOTE: Know this formula, and use it for factoring a difference
of two squares.]
34.
a2 + b2 is prime (not factorable) [NOTE: Know that a sum of two squares in which
each term has a single variable raised to the exponent 2 is not factorable unless it has a
common factor. If it has a common factor, factor out the GCF. Example of common
factor: In 18x2 + 12y2, factor out the GCF, which is 6, to get 6(3x2 + 2y2).]
35.
a2 – 2ab + b2 = (a – b)2 [NOTE: Know this formula, and use it for factoring a perfect
square trinomial with a negative middle term.]
36.
a2 + 2ab + b2 = (a + b)2 [NOTE: Know this formula, and use it for factoring a perfect
square trinomial with a positive middle term.]
37.
3ab2(3b – 2a)
38.
2a(5a2 + 3ab – 1)
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 3 (New Material From: 6.1-6.3, 7.1-7.3, and 10.1)
ANSWERS:
39.
7x3y2(3z4 – 4x2y)
40.
(4x – y)(3x – 2)
41.
(x + y)(5x – 1)
42.
(x – 4)(x – 5)
43.
Prime (Not Factorable) [NOTE: (x – 14)(x – 1) = x2 – 15x + 14, not x2 – 15x – 14]
44.
2(y + 6)(y – 4)
45.
(2x + 3)(x + 2)
46.
(2y + 5)(3y – 4)
47.
x(2x + 5)(x – 6)
48.
(3x – 1)(x – 5)
49.
(3a + 7)2
50.
(y – 6)2
51.
(5x – 1)2
52.
2x(x + 3)(x – 3)
53.
(7a + 9b)(7a – 9b)
54.
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.]
55.
b = 9, or b = –9
56.
x = 7, or x = –7
57.
a = 6, or a = –6
58.
w = 8, or w = –2
59.
y = 6, or y = –2
60.
x = 3, or x = –5
61.
a = 5, or a = 2
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 3 (New Material From: 6.1-6.3, 7.1-7.3, and 10.1)
ANSWERS:
62.
m=
1 + 13
1 − 13
≈ 1.151, or m =
≈ −0.651
4
4
63.
n=
4
, or n = −2
3
64.
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.
65.
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.
66.
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.
67.
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.
68.
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.
69.
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. Please also
remember how to square a binomial such as (x + 7)2 = (x + 7)(x + 7). Multiply all terms
together and combine like terms to get x2 + 14x + 49.
70.
The length of the other leg is 10.5 ft and the length of the hypotenuse is 17.5 ft.
To set up this problem, let x = the length of the other leg. Then x + 7 = the length of the
hypotenuse. Use the Pythagorean Theorem to write the equation x2 + 142 = (x + 7)2.
71.
The lengths of the pieces of rope are 28.5 m and 85.5 m.
To set up this problem, let x = the length of the short piece. Then 3x = 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 + 3x = 114.