Download Algebra 2 Review

Name: ______________________
Class: _________________
Date: _________
ID: A
Algebra 2 Review
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Determine which binomial is not a factor of 4x 4 − 21x 3 − 46x 2 + 219x + 180.
a. x + 4
c. x – 5
b. x + 3
d. 4x + 3
____
2. Determine which binomial is a factor of −x 3 + 6x 2 − 5x − 6.
a. x – 6
b. x + 2
c. x – 5
d.
x–2
Short Answer
3. Use a graphing calculator to find a polynomial function to model the data.
x
1
2
3
4
5
6
7
8
9
10
f(x)
12
4
5
13
9
16
19
16
24
43
4. The table shows the number of hybrid cottonwood trees planted in tree farms in Oregon since 1995. Find
a cubic function to model the data and use it to estimate the number of cottonwoods planted in 2006.
Years since 1995
1
3
5
7
9
Trees planted (in thousands)
1.3
18.3
70.5
177.1
357.3
5. The table shows the number of llamas born on llama ranches worldwide since 1988. Find a cubic function
to model the data and use it to estimate the number of births in 1999.
Years since 1988
Llamas born (in thousands)
1
3
5
7
9
1.6
20
79.2
203.2
416
6. Write the expression (x – 2)(x + 4) as a polynomial in standard form.
7. Miguel is designing shipping boxes that are rectangular prisms. One shape of box with height h in feet,
has a volume defined by the function V(h) = h(h − 5)(h − 9). Graph the function. What is the
maximum volume for the domain 0 < h < 9? Round to the nearest cubic foot.
8. Use a graphing calculator to find the relative minimum, relative maximum, and zeros of
y = 3x 3 + 15x 2 − 12x − 60. If necessary, round to the nearest hundredth.
9. Find the zeros of y = x(x + 2)(x + 3). Then graph the equation.
10. Write a polynomial function in standard form with zeros at –3, 5, and 1.
1
Name: ______________________
ID: A
11. Divide 4x 3 − x 2 + x + 4 by x – 3.
Divide using synthetic division.
12. (x 4 − x 3 − x 2 + 68x − 32) ÷ (x + 4)
13. (x 3 + 4 − 11x + 3x 2 ) ÷ (6 + x)
14. Use synthetic division to find P(–1) for P(x) = x 4 + x 3 + 8x 2 + 10x − 5.
Solve the equation by graphing.
15. x 2 + 3x + 23 = 0
16. 2x 3 − x 2 − 20x = 0
17. 6x = 9 + x 2
Factor the expression.
18. x 3 + 125
19. c 3 − 512
20. x 4 − 52x 2 + 576
21. Solve x 3 + 343 = 0. Find all complex roots.
22. Solve x 4 − 45x 2 = −324.
23. Use the Rational Root Theorem to list all possible rational roots of the polynomial equation
x 3 + 2x 2 + x + 9 = 0. Do not find the actual roots.
24. Find the rational roots of x 4 + 8x 3 + 7x 2 − 40x − 60 = 0.
Find the roots of the polynomial equation.
25. x 3 − 2x 2 + 10x + 136 = 0
26. 2x 3 + 2x 2 − 19x + 20 = 0
27. x 4 − 5x 3 + 11x 2 − 25x + 30 = 0
28. A polynomial equation with rational coefficients has the roots 2 +
roots.
2
7, 7 −
2 . Find two additional
Name: ______________________
ID: A
29. Find a third-degree polynomial equation with rational coefficients that has roots –6 and 3 + i.
30. Find a quadratic equation with roots –1 + 4i and –1 – 4i.
31. Find all zeros of 2x 4 − 5x 3 + 53x 2 − 125x + 75 = 0.
32. The table shows the population of Rockerville over a twenty-five year period. Let 0 represent 1975.
Population of Rockerville
Year
Population
1975
336
1980
350
1985
359
1990
366
1995
373
2000
395
a. Find a quadratic model for the data.
b. Find a cubic model for the data.
c. Graph each model. Compare the quadratic model and cubic model to determine which is a better fit.
Essay
33. Find the rational roots of 4x 3 − 3x − 1 = 0. Explain the process you use and show your work.
Other
34. What are multiple zeros? Explain how you can tell if a function has multiple zeros.
35. Use division to prove that x = 3 is a real zero of y = −x 3 + 9x 2 − 38x + 60.
36. A polynomial equation with rational coefficients has the roots
additional roots and name them.
3
7 and −
3 . Explain how to find two
ID: A
Algebra 2 Review
Answer Section
MULTIPLE CHOICE
1. A
2. D
SHORT ANSWER
3.
4.
5.
6.
7.
8.
f(x) = 0.08x4 – 1.73x3 + 12.67x2 – 34.68x + 35.58
T(x) = 0.4x 3 + 0.8x 2 + 0.1x; 630.3 thousand trees
L(x) = 0.5x 3 + 0.6x 2 + 0.3x + 0.2; 741,600 llamas
x2 + 2x – 8
42 ft3
relative minimum: (0.36, –62.24), relative maximum: (–3.69, 37.79),
zeros: x = –5, –2, 2
9. 0, –2, –3
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
f(x) = x 3 − 3x 2 − 13x + 15
4x 2 + 11x + 34, R 106
x 3 − 5x 2 + 19x − 8
x 2 − 3x + 7, R –38
–7
no solution
0, 3.42, –2.92
3
(x + 5)(x 2 − 5x + 25)
(c − 8)(c 2 + 8c + 64)
(x − 4)(x + 4)(x − 6)(x + 6)
1
ID: A
21. −7,
7 ± 7i
3
2
6, –6, 3, –3
–9, –3, –1, 1, 3, 9
–6, –2
3 ± 5i, –4
3 +i 3 −i
26.
,
, −4
2
2
22.
23.
24.
25.
27. 2, 3, ± i
5
28. 2 − 7 , 7 + 2
29. x 3 − 26x + 60 = 0
30. x 2 + 2x + 17 = 0
3
31. 1, , ± 5i
2
32. a. y = 0.023x 2 + 1.549x + 338.571
b. y = 0.0079x 3 − 0.2716x 2 + 4.2378x + 335.6270
c.
The cubic model is a better fit.
2
ID: A
ESSAY
33.
[4]
[3]
[2]
[1]
Step 1:
List the possible rational roots by using the Rational Root Theorem. The leading
coefficient is 4 with factors of ±1, ±2, and ±4. The constant term is –1 with factors
factor of −1
of –1 and 1. The only possible roots of the equation have the form
.
factor of 4
1
1
Those roots would be ±1, ± , and ± .
2
4
Step 2:
Test each possible rational root in the equation. The only roots that satisfy the
1
equation are − and 1.
2
an error in computation or missing part of the explanation
several errors in computation or in the explanation
one root given with no explanation
OTHER
34. If a linear factor of a polynomial is repeated, then the zero is repeated and the function has multiple
zeros. To determine whether a function has a multiple zero, factor the polynomial. If a factor is repeated
in the factored expression, then it is a multiple zero.
35. −x 3 + 9x 2 − 38x + 60 ÷ (x – 3) = −x 2 + 6x − 20 with no remainder, so x = 3 is a real zero of the
function.
36. By the Irrational Root Theorem, if
root, then its conjugate
7 is a root, then its conjugate –
3 is also a root. Two additional roots are –
3
7 is also a root. If −
7 and
3.
3 is a