Name: ________________________ Class: ___________________ Date: __________
ID: R
Secondary Math 3 Honors - Polynomial and Polynomial Functions Test Review
1. Write –3x2(–2x2 – 5x3) in standard form.
State whether the function is even, odd, or
neither. Show your work. Answer the questions
using an algebraic method. Check by graphing.
8. f(x) = |x + 1 | + |x − 1 |
2. The polynomial x 3 + 6x 2 − 55x − 252 expresses the
volume, in cubic inches, of a shipping box, and the
width is (x + 4 ) in. If the width of the box is 16 in.,
what are the other two dimensions? (Hint: The
height is greater than the depth.)
Height: __________
answer: ____________
.
9. f(x) = x 3 + 2x − 4
answer: _____________
Depth:__________
3. State the number of possible real zeros and turning
points of f(x) = x 6 + 4x 5 − 5x 4 . Then determine all
of the real zeros by factoring.
.
10. Is the function graphed below even, odd, or
neither?
Number of possible real zero: ______
Number of possible turning points: ______
Real Zeros: _________
Simplify the given expression.
Ê
ˆ
4. (3x − 5) ÁÁ 2x 2 + 6x − 3 ˜˜
Ë
¯
answer: ____________
11. True or False. A polynomial can have negative
exponents as powers of the variables.
Ê
ˆ Ê
ˆ
5. ÁÁ 8x 2 − 14x − 25 ˜˜ + ÁÁ −5x 3 + 11x 2 − 2 ˜˜
Ë
¯ Ë
¯
12. True or False. When two polynomials are
subtracted the result is sometimes another
polynomial.
Ê
ˆ Ê
ˆ
6. ÁÁ −8x 2 − 2x + 13 ˜˜ − ÁÁ 13x 2 + 16x − 2 ˜˜
Ë
¯ Ë
¯
Find the real solutions of the equation by
graphing. Round to 3 decimal places if
necessary.
7. Divide using method of your choice.
−2x 3 − 4x 2 + 4x − 3 by x – 4.
13. x 3 + 2x + 2 = 0
1
Name: ________________________
ID: R
What are the real or imaginary solutions of each
polynomial equation?
Factor the polynomial completely.
18. 7x 2 − 17x + 10
3
2
14. 2x − 3x − 16x + 24 = 0
.
15. x 4 − 13x 2 = −36
19. 24x 3 − 40x 2 + 15x − 25
16. 125x 3 + 343 = 0
20. 3x3 – 24x2 + 45x
17. 20x 2 − 11x − 4 = 0
What is the degree of the polynomial that generates the given data?
21.
x
–6
–2
0
2
6
y
138
14
0
18
150
a.
b.
linear model
cubic model
c.
d.
quadratic model
none of these
23. Describe the end behavior of the graph of f(x) =
x 2 (x + 10)(–2x + 4) using limits.
What are the relative maximum and minimum
of the function? Use interval notation to
describe where the function is increasing or
decreasing. State the domain and range.
22. f(x) = 4x 3 + x 2 − 3x
Minimum: ________
a.
b.
Maximum: ________
c.
Increasing: _________ Decreasing: _______
d.
Domain: _________ Range: _________
2
As x → − ∞, f(x) → +∞
As x→ + ∞, f(x)→ -∞
As x → − ∞, f(x) → -∞
As x→ + ∞, f(x)→ -∞
As x → − ∞, f(x) → -∞
As x→ + ∞, f(x)→ +∞
As x → − ∞, f(x) → +∞
As x→ + ∞, f(x)→ +∞
Name: ________________________
ID: R
24. Use the remainder theorem to find P(–4) for
P(x) = x 4 + 8x 3 − 10x 2 + x + 6 .
29. Graph y = 10x 2 − 9 − x 4 . How many turning points
are there? State the domain and range.
Divide using long division.
Ê
ˆ
25. ÁÁ −8x 6 + 10x 5 + x 4 + 7x 3 − 18x 2 + 30x − 21 ˜˜ ÷
Ë
¯
(−2x + 3)
Turning points: __________
Given a polynomial and one of its factors, find
the remaining factors of the polynomial. Write
the polynomial in factored form. Don’t forget
the given factor.
Domain : ___________
Range: _____________
26. 12x 3 − 100x 2 + 168x + 30; x − 6
For the given graph,
a. describe the end behavior,
b. determine whether it represents an
odd-degree or even-degree polynomial
function, and
c. state the number of real zeros.
27. 4x 3 − 48x 2 − 49x + 588; x − 12
30.
28. The perimeter of a rectangle is 24a + 4b The length
of rectangle is 9a − 2b. Find the measure of the
width.
a. __________________________
b.___________ c. ____________
3
Name: ________________________
ID: R
31.
A manufacturer produces boxes for a calculator company. The boxes have a volume of 240cm3 . The height of
each box is 6cm less than the width. The length is 1 cm less than twice the width. Find the dimensions.
Divide using synthetic division.
35. Determine the sign of the leading coefficient and
the least possible degree of the polynomial for the
given graph.
32. Divide 20x 3 + 150 − 5x by (x + 2 ).
33. What is a cubic polynomial function in standard
form with zeros –4, 1, and –5?
36. For the following function, find the intervals where
the function is increasing or decreasing.
f(x) = x 3 − 4x 2 + x + 8
34. Determine the zeros and the end behavior of f(x) =
−x(x − 1)(x − 3) 2 . Then sketch a graph of the
function.
37. Is x + 4 a factor of f(x) = x 3 + 12x 2 + 29x − 7
.
Zeros: _________________
End behavior: ________________________
________________________
4
ID: R
Secondary Math 3 Honors - Polynomial and Polynomial Functions Test Review
Answer Section
1. 15x5 + 6x4
2. height: 21 in.
depth: 5 in.
3. 6 real zeros and 5 turning points; 0, 1, and –5
4. 6x 3 + 8x 2 − 39x + 15
5. −5x 3 + 19x 2 − 14x − 27
−21x 2 − 18x + 15
−2x 2 − 12x − 44, R –179
even
neither
neither
F
F
–0.771
3
14.
,±2 2
2
15. 2, –2, 3, –3
6.
7.
8.
9.
10.
11.
12.
13.
7 35 ± 35i 3
16. − ,
5
50
4 1
17.
,−
5 4
18. (7x − 10)(x − 1)
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
(8x 2 + 5)(3x − 5)
3x(x – 5)(x – 3)
C
The relative maximum is at (–0.59, 1.3) and the relative minimum is at (0.42, –0.79).
Increasing: ÊÁË −∞,−0.59 ˆ˜¯ and ÊÁË 0.42,∞ ˆ˜¯ Decreasing : ÊÁË −0.59,0.42 ˆ˜¯
Domain: ÊÁË −∞,∞ ˆ˜¯ Range: ÊÁË −∞,∞ ˆ˜¯
B
–414
−3
4x 5 + x 4 + x 3 − 2x 2 + 6x − 6 +
−2x + 3
(6x + 1)(2x − 5)(x − 6)
(2x −7)(2x + 7)(x − 12 )
3a + 4b
The measure of the width of the rectangle is given by the expression (24a + 4b) − 2 (9a − 2b ) Simplify this
expression by grouping similar terms and then combining them.
1
ID: R
29.
There are three turning points.
30. The end behavior of the graph is f (x ) → −∞ as x → +∞ and f (x ) → −∞ as x → −∞ .
It is an even-degree polynomial function.
The function has four real zeros.
31. 15 cm. by 8 cm. by 2 cm.
32. 20x 2 − 40x + 75
33. f(x) = x 3 + 8x 2 + 11x − 20
34. 0, 1, 3 (multiplicity of 2)
As x → − ∞, f(x) → -∞
As x→ + ∞, f(x)→ −∞
35. negative, degree 3
36. increasing ÊÁË −∞,0.1 ˆ˜¯ ∪ ÊÁË 2.5,∞ ˆ˜¯
decreasing (0.1, 2.5)
37. No. (use synthetic division to determine if it is a factor.)
2