Practice Exercises for Exam 3
Sections 5.5-8.3 (excluding 6.4, 6.6, 6.7)
Math 1010 – Intermediate Algebra – Fall 2011
If you would like partial credit, then you must SHOW YOUR WORK,
carefully documented. No magical answers.
1. Factor the trinomial.
(a) 36x2 − 60xy + 25y 2
(b) 3m3 − 18m2 + 27m
(c) x2 − 5x + 6
(d) y 2 + 7y − 30
(e) t2 − 6t − 16
(f) x2 + 7x + 10
(g) 6x2 − 5x − 25
(h) 10y 2 − 7y − 12
(i) 15x2 + 4x − 3
(j) 10t3 + 2t2 − 36t
2. Solve the equation.
(a) 5y − y 2 = 0
(b) 2x2 = 32x
(c) 3y 2 − 48 = 0
(d) x2 − 10x + 24 = 0
(e) x2 − x − 12 = 0
(f) x2 + 42 = 13x
(g) 11 + 32y − 3y 2 = 0
(h) x2 + 16x + 57 = −7
(i) x2 − 12x + 21 = −15
(j) x(x + 2) − 10(x + 2) = 0
(k) x(x − 15) + 3(x − 15) = 0
3. The rectangular floor of a storage shed has an area of 540 square feet. The length of the floor is
7 feet more than its width. Find the dimensions of the floor.
4. The height of a triangle is 2 inches less than its base. the area of the triangle is 60 square
inches. Find the base and height of the triangle.
5. Find the domain.
2x
+1
x − 12
(b) f (x) =
x(x2 − 16)
(a) f (x) =
x2
6. Simplify the rational expression.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
6x4 y 2
15xy 2
2(y 3 z)2
28(yz 2 )2
5b − 15
30b − 120
x+3
2
x − x − 12
x2 − 5x
2x2 − 50
15(x2 y)3 12y
·
3y 3
x
2
x − 16
3
· 2
6
x − 8x + 16
xy
25y 2 ÷
5
x2 + 3x + 2
3x2 + x − 2
x2 − 7x x2 − 14x + 49
÷
x+1
x2 − 1
4x
3x − 7
9
+
−
x+2
x+2
x+2
3
4
+
5x2 10x
1
3
+
x + 5 x − 12
6
4x + 7
− 2
x − 5 x − x − 20
5
25 − x
+ 2
x + 2 x − 3x − 10
7. Perform the division. (Long division)
4x4 − x3 − 7x2 + 18x
x−2
4
2
x − 3x + 2
(b)
x2 − 1
x4 − 4x3 + 3x
(c)
x2 − 1
(a)
8. Rewrite the expression using rational exponents.
√
3
(a) z z 2
√
4
x3
(b) √
x4
√
3
(c) a3 b2
�
4 √
(d)
x
(e)
(3x + 2)2/3
√
3
3x + 2
9. Describe the domain of the function.
√
(a) f (x) = 9 − 2x
√
(b) g(x) = 3 x + 2
10. Simplify the radical expressions.
�
(a)
24x3 y 4
√
(b) 0.16s6 t3
√
4
(c) 48u4 v 6
11. Rationalize the denominator and simplify.
�
5
(a)
6
4y
(b) √
10z
�
3 16t
(c)
s2
12. Combine the rational expressions, if possible.
√
√
√
(a) 2 24 + 7 6 − 54
√
√
√
√
(b) 5 x − 3 x + 9 x − 8 3 x
�
�
(c) 2x 3 24x2 y − 3 3x5 y
13. Multiply and Simplify.
√
√
(a) 15 · 20
√ √
√
(b) 10( 2 + 5)
√
√ √
√
(c) ( 3 − x)( 3 + x)
√
(d) (4 − 3 2)2
14. Rationalize the denominator. Because I said so.
√
2−1
(a) √
3−4
√
2 + 20
√
(b)
3+ 5
15. Solve the equation and check your solution(s).
√
(a) 2x − 8 = 0
√
(b) 4x + 6 = 9
√
(c) 3 5x − 7 − 3 = −1
√
(d) y − 2 = y + 4
√
√
(e) 1 + 6x = 2 − 6x
16. Determine the length and width of a rectangle with a perimeter of 46 inches and a diagonal of
17 inches.
√
17. The velocity of a free-falling object can be determined from the equation v = 2gh, where v is
the velocity in feet per second, g = 32 per second per second, and h is the distance (in feet) the
object has fallen.
(a) Find the height from which a brick has been dropped when it strikes the ground with a
velocity of 64 feet per second.
(b) Find the height from which a wrench has been dropped when it strikes the ground with a
velocity of 112 feet per second.
(c) Do the above questions get you pumped about mathematics because they are about the
“real world”?
18. Write the number in i-form.
√
(a) −48
√
(b) −0.16
√
(c) 10 − 3 −27
�
3
(d) 34 − 5 − 25
19. Perform the operation and write the result in standard form.
(a) (−4 + 5i) − (−12 + 8i)
(b) (4 − 3i)(4 + 3i)
7
(c)
3i
−3i
(d)
4 − 6i
3 − 5i
(e)
6+i
20. Solve the quadratic equations any way you want to (factoring, completing the square, quadratic
formula), but make sure that you practice all three different methods. For four of these
exercises, you will want to make a clever substitution to create a quadratic equation.
(a) 3y 2 − 27 = 0
(b) 4y 2 + 20y + 25 = 0
(c) 2x2 − 2x − 180 = 0
(d) 9x2 + 18x − 135 = 0
(e) 10x − 8 = 3x2 − 9x + 12
(f) z 2 = 144
(g) 2x2 = 98
(h) (x − 16)2 = 400
(i) (x + 3)2 = 900
(j) z 2 = −121
(k) y 2 + 50 = 0
(l) (y + 4)2 + 18 = 0
(m) (x − 2)2 + 24 = 0
(n) x4 − 4x2 − 5 = 0
√
(o) x − 4 x + 3 = 0
(p) (x2 − 2x)2 − 4(x2 − 2x) − 5 = 0
(q) x2/3 + 3x1/3 − 28 = 0
(r) x2 − 6x − 3 = 0
(s) x2 + 12x + 6 = 0
(t) v 2 + 5v + 4 = 0
(u) u2 − 5u + 6 = 0
(v) y 2 − 23 y + 2 = 0
(w) v 2 + v − 42 = 0
(x) 2y 2 + y − 21 = 0
(y) 2x2 − 3x − 20 = 0
(z) 3x2 + 12x + 4 = 0
ANSWER KEY
1. (a) (6x − 5y)2
(b) 3m(m − 3)2
(c) (x − 3)(x − 2)
(d) (y + 10)(y − 3)
(e) (t − 8)(t + 2)
(f) (x + 2)(x + 5)
(g) (3x + 5)(2x − 5)
(h) (5y + 4)(2y − 3)
(i) (5x + 3)(3x − 1)
(j) 2t(5t − 9)(t + 2)
2. (a) 0,5
(b) 0,16
(c) 4, -4
(d) 4,6
(e) -3, 4
(f) 6,7
1
(g) − , 11
3
(h) -8
(i) 6
(j) -2, 10
(k) -3, 15
3. 20 feet × 27 feet
4. Base: 12 inches; Height: 10 inches
5. (a) (−∞, ∞)
(b) (−∞, −4) ∪ (−4, 0) ∪ (0, 4) ∪ (4, ∞)
6. (a)
(b)
(c)
(d)
(e)
2x3
, x �= 0, y �= 0
5
y4
, y �= 0
14z 2
b−3
6(b − 4)
1
, x �= −3
x−4
x
, x �= 5
2(x + 5)
(f) 60x5 y, x �= 0, y �= 0
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
x+4
2(x − 4)
125y
, y �= 0
x
1
, x �= −2, x �= −1
3x − 2
x(x − 1)
, x =�= −1, x �= 1
x−7
7x − 16
x+2
2x + 3
5x2
4x + 3
(x + 5)(x − 12)
2x + 17
(x − 5)(x + 4)
4x
(x + 2)(x − 5)
7. (a) 4x3 + 7x2 + 7x + 32 +
(b) x2 − 2, x �= ±1
(c) x2 − 4x + 1 −
1
x+1 , x
8. (a) z 5/3
1
(b) 5/4
x
(c) ab2/3
(d) x1/8
(e) (3x + 2)1/3 , x �= − 23
9. (a) (−∞, 92 ]
10.
11.
12.
13.
(b) (−∞, ∞)
√
(a) 2xy 2 6x
√
(b) 0.4|s3 |t t
√
4
(c) 2|uv| 3v 2
√
30
(a)
6
√
2y 10z
(b)
5z
√
3
2 2st
(c)
s
√
(a) 8 6
√
√
(b) 14 x − 9 3 x
�
(c) 3x 3 3x2 y
√
(a) 10 3
64
x−2
�= 1
√
√
(b) 2 5 + 5 2
(c) 3 − x
√
(d) 34 − 24 2
√
√
√
6+4 2− 3−4
14. (a) −
13
√
(b) 5 − 1
15. (a) 32
(b) 9/4
(c) 3
(d) 5
(e) 3/32
16. 8 inches by 15 inches
17. (a) 64 feet
(b) 196 feet
(c) No.
√
18. (a) 4 3i
(b) 0.4i
√
(c) 10 − 9 3i
√
(d) 34 − 3i
19. (a) 8 − 3i
(b) 25
(c) − 73 i
(d)
(e)
9
26
13
37
−
−
3
13 i
33
37 i
20. (a) 3,-3
(b) -5/2
(c) -9, 10
(d) -5,3
(e) 4/3, 5
(f) 12, -12
(g) 7, -7
(h) -4, 36
(i) -33, 27
(j) 11i, −11i
√
√
(k) 5 2i, −5 2i
√
(l) −4 ± 3 2i
√
(m) 2 ± 2 6i
√
(n) ± 5, ±i
(o) 1,9
√
√
(p) 1,1+ 6, 1 − 6
(q) -343, 64
√
(r) 3 ± 2 3
√
(s) −6 ± 30
(t) -4,-1
(u) 2,3
(v)
1
3
±
√
17
3 i
(w) -7,6
(x) -7/2, 3
(y) -5/2, 4
√
2 6
(z) −2 ±
3