Math 1010 Section 5
William Malone
Review 2
Name:
This review will be graded on completion and is due on December 2nd. Complete
solutions can be found on my website.
1. Use the rules of exponents to simplify the following
(3x2 y 3 )4
.
(6x−3 y 2 )5
2. Add the following polynomials together. 3x − 2 + x3 and 4x2 − 5x3 + 4x − 2
3. Multiply and simplify (3x − 2 + x3 )(4x2 − 2x + 5x3 ).
4. Factor completely −20x8 + 160x5 + 5x10 − 40x7 .
5. Factor completely 10x4 − 29x2 + 12 (Factors into 4 linear terms).
6. Find all solutions to the equation −2x2 + 5x + 19 = 4x2 − 4x + 4
7. Graph the domain of the function on the real number line.
√
√
100
13
x2 − 1
3x2 − 2x + 1
−
x2 − 10x + 24
x2 − 5x + 6
8. Simplify the following to a rational expression.
x2
x+1
x−1
− 2
− 10x + 21 x − 5x + 14
9. Simplify the following to a rational expression
3x−1
2x+1
5x−6
x+2
10. Perform the division
x5 − 3x4 + 3x3 + 7x2 − 2x
x2 − 3x + 5
11. Perform the division
3x4 + 24x3 − 40x2 − 30x + 36
x+6
12. Find all solutions to the equation
x+7 x−6
−
=4
x−3 x+5
3
13. Simplify the following to an integer
(7 5 )10
.
(710 7−8 )2
14. Simplify
√
x+1−
√
x − 1.
15. Simplify the following into a radical of a single polynomial.
16. Simplify the following
√
√
x + 1 6 x + 1.
√ √
√2−√3 .
6+ 8
√
3
2x − 1 = x + 1.
√
√
18. Find all solutions to the equation x − 1 − 2x + 1 = 2.
17. Find all solutions to the equation
19. Simplify the following to a complex number
1 − i 4 − 3i
−
.
2 + i 3 − 2i
20. Find all solutions to the equation (x + 7)118 = 10.
√
21. Find all solutions to the equation x − 4 x + 1 = 0.
22. Sketch the graph of the following function using shifts and reflections of known
graphs. f (x) = −x2 + 10x − 30
23. Graph on the real number line all solutions to the inequality x3 −8x+x2 −8 > 0
24. Graph on the real number line all solutions to
−x4 (x−1)7 (x+9)5
(x+1)4 (x−2)2 (x+4)3 (x−5)5
>0
25. Graph the function f (x) = −5−x−5 + 6 using shifts and reflections of graphs.
26. Graph the function g(x) = − log3 (−x + 6) + 2 using shifts and reflections of
graphs.
27. Graph the domain of the function T (x) = log5 (x + 6) + log7 (x − 2)
√
28. Let f (x) = x2 + 1 and g(x) = 3x + log5 (x) then find (f ◦ g)(x) and (g ◦ f )(x)
29. Let T (x) = −3−x+6 − 10. Write T (x) as a composition of 2, 3, and 4 different
functions.
1
30. Simplify the following expression to a rational number log4 ( 256
)
31. Use the properties of logarithms to expand the following expression.
√
log2 (x3 y 5 z 5 10 w)
32. Use the properties of logarithms to collapse the following expression.
log2 (x + 6) −
1
log4 (x − 2) + 4 log8 (x + 1) − 10 log16 (x)
2
33. Find all solutions to the equation 4 · 5x+6 = 10.
34. Find all solutions to the equation log9 (x + 5) = 10.