Download Math 1110 Test 1. Fall 2012

Math 1110 Test 1. Fall 2012
Problems 1–4. Simplify. You final answer should
not contain any negative exponents.
1.
(2x2 y −1 z)3
2.
3.
3
4.
2x4 y 7 z
−2
x y 3 z −2
9x2 y ·
8x3
y −6
3
2
3x10
y4
2/3
Problems 5–6. Simplify the following rational
expressions.
5.
1
1
−
x2 + 3x + 2 x2 − 2x − 3
6.
x
−
y
1
−
x2
y
x
1
y2
Problems 7–11. Factor each expression completely.
7.
x2 − 4x − 12
8.
2x2 + 7x + 5
9.
x3 + 27
10.
x4 − 16
11.
x3 + 3x2 − x − 3
Problems 12–15. Solve each of the following
equations.
12.
3x2 − 11x = 4
13.
x=
14.
5
28
x+5
=
+ 2
x−2
x+2 x −4
15.
x − 6x1/2 + 8 = 0
√
2x − 2 + 1
16. Use the quadratic formula to find all real
solutions of the equation 2x2 + 5x = 1.
solutions
6.
x
−
y
1
−
x2
1.
(2x2 y −1 z)3 = 23 x6 y −3 z 3 =
8x6 z 3
.
y3
x3 y − xy 3
y 2 − x2
2.
2x4 y 7 z
x−2 y 3 z −2
y
x x2 y 2
·
1 x2 y 2
y2
2
=
22 x8 y 14 z 2
= 4x12 y 8 z 6 .
x−4 y 6 z −4
xy(x2 − y 2 )
(y 2 − x2 )
= −xy
3.
3
9x2 y·
3
3x10
=
y4
=
4.
8x3
y −6
3
2/3
3
9x2 y · 3x10
y4
3x4
27x12
=
y3
y
82/3 x2
= −4 = 4x2 y 4
y
7.
8.
9.
x2 − 4x − 12 = (x − 6)(x + 2)
2x2 + 7x + 5 = (2x + 5)(x + 1)
x3 + 27 = (x + 3)(x2 − 3x + 9)
10.
5.
x4 − 16
1
1
− 2
2
x + 3x + 2 x − 2x − 3
= (x2 − 4)(x2 + 4)
= (x − 2)(x + 2)(x2 + 4)
1
1
−
=
(x + 1)(x + 2) (x + 1)(x − 3)
=
(x − 3) − (x + 2)
(x + 1)(x + 2)(x − 3)
=
−5
(x + 1)(x + 2)(x − 3)
11.
x3 + 3x2 − x − 3
= x2 (x + 3) − 1(x + 3)
= (x + 3)(x2 − 1)
= (x + 3)(x + 1)(x − 1)
12.
3x2 − 11x − 4 = 0
(3x + 1)(x − 4) = 0
3x + 1 = 0,
x = −1/3,
x−4=0
x=4
13.
x−1=
√
2x − 2
(x − 1)2 = 2x − 2
x2 − 2x + 1 = 2x − 2
x2 − 4x + 3 = 0
(x − 3)(x − 1) = 0
x = 3,
x=1
Check to see that both 1 and 3 are valid solutions by plugging into the original equation.
They both work.
14. Multiply both sides of the equation by the
common denominator (x + 2)(x − 2):
(x + 2)(x + 5) = 5(x − 2) + 28
x2 + 7x + 10 = 5x − 10 + 28
x2 + 2x − 8 = 0
(x + 4)(x − 2) = 0
x = −4,
x=2
Note that 2 is not a solution, however, because
it would cause a zero in a denominator of a
fraction.
15.
(x1/2 )2 − 6x1/2 + 8 = 0.
Put u = x1/2 . Then
u2 − 6u + 8 = 0
(u − 2)(u − 4) = 0
u = 2,
u=4
x1/2 = 2,
x = 4,
x1/2 = 4
x = 16
16.
x=
−5 ±
√
25 − 4 · 2 · (−1)
−5 ± 33
=
.
2·2
4