0
0.1
Miscellaneous
Polynomial Roots and Factorization (6.6)
1. Find all the roots of the polynomial (x − 2)(x + 1)(3x − 5)
5
Answer: −1, 2, and
3
2. Find all the roots of the polynomial 3(x − 5)(2x + 7)
7
Answer: 5 and −
2
3. Find all the roots of the polynomial x2 (3x + 1)(6 − 2x)
1
Answer: 0, − , and 3
3
4. Find all the roots of the polynomial 4(x − 5)2 (2x + 10)
Answer: 5 and −5
5. Find all the roots of the polynomial (x + 3)(x − 7)(x − 9)
Answer: −3, 7, and 9
6. Find all the roots of the polynomial 5(3x + 1)2 (9 − 5x)
9
1
Answer: − and
3
5
7. Find a polynomial that has roots −2 and 5.
Answer: (x + 2)(x − 5)
8. Find a polynomial that has roots 0, −1 and 7.
Answer: x(x + 1)(x − 7)
9. Find a polynomial that has roots −3, −5 and 17
Answer: (x + 3)(x + 5)(x − 17)
10. Find a polynomial that has a root at 4
Answer: x − 4
1
11. Find a polynomial that has roots 3 and 5 and the coefficient of the
leading term is −2
Answer: 2(x − 3)(x − 5)
12. Find a polynomial of degree 3 that has roots 2 and 0.
Answer: x2 (x − 2)
0.2 Solving Quadratic Equations by the Quadratic
Formula (9.3)
13. Put each of the following quadratic equations in standard form and
identify the coefficients a, b, and c.
(a) 3x2 − 5x + 10 = −14
(b) x2 + 5(9 − x) = 12 − 2x2
(c) 4x2 + 5x − 9 = 6x2 − 3x − 8
14. Solve each of the following quadratic equations using the quadratic
formula
(a) x2 − 7x + 12 = 0
(b) 3x2 − 2x = 10
(c) 4x2 − x + 5 = 2x2 + 3x − 4
(d) 5x2 + 4x + 12 = 0
0.3 Roots and Fractional Exponents (see section
8.1)
15. Simplify the following as much as possible.
√
(a) 81 = 9
√ 2
(b)
39 = 39
√
2
(c)
x3 + 4x = x3 + 4x
2
(d)
√
a2 + 4
16. Use a calculator to estimate the following to three decimal places
√
(a) 17 ≈ 4.12311
√
(b) − 43.2 ≈ −6.57267
17. Simplify the following as much as possible.
√
4
(a) 81 = 3
√ 2
4
(b) − 625 = 25
√ 2
3
64 = 16
(c)
√
(d) 3 −64 = −4
18. The image below is a schematic of someone looking to the horizon.
The circle represents the Earth; the point O represents the center of
the Earth; the small, slightly thickened segment represents the person.
It is a fact of geometry that the angle at point A is a right angle (you
can take this for granted). If the person is 6 feet tall, how far can they
see looking to the horizon? (NOTE: you’ll need to know the radius of
the Earth–google it) What if you are standing in a tower that is 100
feet tall?
3
A
O
0.4 Absolute Value and the Number Line (1.3,
2.7)
19. Plot the points 4, −4, 7/3, 2
-4
3
and −2.4 on the number line below.
4
0
-2.4
73
3
2
4
4
20. If a < b < c < d, label the marked points on the number line below
using the letters a, b, c, and d.
a
b
4
0
c
d
21. Mark each of the following true or false.
(a) −4 < 5
Answer: True
(b) −4 < −5
Answer: False
(c) − |−3| ≥ 0
Answer: False
(d) |7 − 12| > 0
Answer: True
22. Compute the following
(a) |3|
Answer: 3
(b) |−6|
Answer: 6
(c) |4 − 7|
Answer: 3
(d) − |20 − 6|
Answer: −14
23. On the number line below, shade all the values of x that satisfy −1 <
x<4
0
24. On the number line below, shade all the values of x that satisfy |x| < 2
0
5
25. On the number line below, shade all the values of x that satisfy |x| ≥ 2
0
26. Solve and graph the inequality |3x − 4| ≤ 2
2
Answer: The solution set is ≤ x ≤ 2
3
0
27. Solve and graph the inequality |2x + 7| > 4
11
3
Answer: The solution set is x < −
together with x > −
2
2
0
28. Solve and graph the inequality |4 − x| ≥ 5
Answer: The solution set is {x : x ≤ −1 or x ≥ 9}
0
29. Solve and graph the inequality |2x − 4| < 19
23
15
Answer: {x : − < x < }
2
2
0
6
30. Solve and graph the inequality |x + 12| ≤ 5
Answer: {x : −17 ≤ x ≤ 17}
0
31. Solve and graph the inequality |18 − 3x| > 10
8
28
or x < }
Answer: {x : x >
3
3
0
1
32. Solve and graph the inequality x − 4 ≥ 6
3
Answer: {x : x ≤ −6 or x ≥ 30}
x-4)¿=6.pdf
0
33. Solve and graph the inequality |5(x − 2)| < 39
29
49
Answer: {x : − < x < }
5
5
0
7