MHF 4U Advanced Functions Grade 12
POLYNOMIAL EQUATIONS
TEST 11
Multiple Choice
Identify the choice that best completes the statement or answers the question.
1. A family of quadratic functions has zeros –3 and 5. Which of the following is a member of this family? a.
𝑦 = −
1
2
( x + 3)( x – 5) b. y = 2( x + 3) 2 ( x – 5) c. y = 4( x + 3)( x – 5)( x + 5) d. y = ( x – 3)( x + 5)
2. Which of the following represents a family of cubic polynomials with zeros
–4, 2, and 6? a. y = ( x + 4)( x – 2)( x – 6) b. y = k ( x + 4)( x – 2)( x – 6) c. y = ( x – 4)( x + 2)( x + 6) d. y = k ( x + 2)( x – 4)( x + 6)

3. Examine the graphs of the following polynomial functions. Which graph does not belong to the same family? a.
c.
b.
d.
4. What is an equation for the cubic function represented by this graph? a.
y = 3( x – 1) 2 ( x + 1) b. y = –3( x – 1) 2 ( x + 1) c. y = 3( x + 1) 2 ( x – 1) d. y = –3( x + 1) 2 ( x – 1)

5. A family of polynomials has equation y = k ( x – 3)( x + 2) 3 . What is the value of k for the family member represented by this graph? a . k = 48 b. k = –48 c. k = 2 d. k = –2
6. How many cubic functions have zeros –10, –5, and 4? a. 1 b. 2 c. 3 d. infinitely many
7. What is an equation for the family of cubic functions with
4 zeros –8, 3, and
3
? a.
y = ( x + 8)( x – 3)(4 x – 3) b. y = ( x + 8)( x – 3)(3 x – 4) c. y = k ( x + 8)( x – 3)(3 x – 4) d. y = k ( x + 8)( x – 3)(4 x – 3)

8. Which statement corresponds to the values of x shown on the following number line? a. −4 ≤ 𝑥 < 2 b. −4 < 𝑥 ≤ 2 c. 𝑥 ≤ −4 𝑜𝑟 𝑥 > 2 d.
−4 ≤ 𝑥 𝑜𝑟 𝑥 > 2
9. Which of the following number lines depicts the solution to
3 x ( x 2 – 4) + 2 x > 3 x 3 + 5( x + 1)?
10. A graph of y = x 4 + x 3 – 7 x 2 – x + 6 is shown. Use the graph to solve the inequality x 4 + x 3 – 7 x 2 – x + 6 ≥ 0. a . −1 ≤ 𝑥 ≤ 1 b. −3 ≤ 𝑥 ≤ −1 c.
−3 ≤ 𝑥 ≤ −1, 1 ≤ 𝑥 ≤ 2 d. 𝑥 ≥ −3, −1 ≤ 𝑥 ≤ 1, 𝑥 ≤ 2

11. A graph of y = 2 x 4 + 4 x 3 – 24 x 2 – 80 x – 64 is shown. Use the graph
to solve the inequality: 2 x 4 + 4 x 3 – 24 x 2 – 80 x – 64 ≥ 0 a. 𝑥 = −2, 4 b. 𝑥 ≤ −2, 𝑥 ≥ 4 c. −2 ≤ 𝑥 ≥ 4 d. −2 ≤ 𝑥 ≤ 4
12. Toby is solving x 3 + x 2 – 5 x < 0 using a graphing calculator. He determines that the zeros of y = x 3 + x 2 – 5 x are 0 and approximately –2.8 and 1.8. Based on this information, what is the solution to x 3 + x 2 – 5 x < 0? a.
𝑥 = 0, 𝑥 = −2, 𝑥 = 1.8
b. 𝑥 < −2.8, 0 < 𝑥 < 1.8
c. −2.8 < 𝑥 < 0, 𝑥 > 1.8
d. −2.8 < 𝑥 < 1.8

13. The inequality x 4 – 5 x 2 + 4 ≥ 0 can be solved by a.
considering all cases b.
graphing c.
d.
using intervals all of the above
14. When solving 3( x – 2)(2 x – 1)( x + 4) ≥ 0 by considering all cases, how many cases must be considered? a.
2 b.
3 c.
4 d.
8
15. Which inequality has x ≤ −4 or 0 ≤ 𝑥 ≤ 3 as its solution? a. x ( x – 3)( x + 4) ≤ 0 b. −2 x ( x – 3)( x + 4) ≥ 0 c. 100 x ( x – 3)( x + 4) ≤ 0 d. all of the above
16. Which of the following is true about solving polynomial inequalities algebraically? a.
When solving by using intervals, cubic inequalities require fewer intervals to be tested than quadratic inequalities.
b.
When solving a quartic inequality by considering cases, eight cases must be considered.
c.
When solving by considering cases or by using intervals the polynomial must be factorable.
d.
When solving by using intervals, the number of zeros of the polynomial is the number of intervals that must be tested.

Completion
Complete each statement.
17. Polynomial functions with the same zeros are said to belong to the same
_______________.
18. When the equal sign in a polynomial equation is replaced with either < ,
≤
, ≥ , or >, then a polynomial _______________ is created.
19. The solution to x 2 ( x 2 + 1) > 0 is _______________.
Short Answer
20. Solve the following inequalities. Round answers to one decimal place. a) x 3 – 7 > 0

b) ( x + 14) 3 ≤ 1
21. Solve the following inequalities using an algebraic method. a) x 2 + x
– 2 > 0

b) x ( x – 2)( x – 1)( x + 1) ≤ 0 c) x 3 ≥ 9 x

d) – x 3 + 2 x 2 + 4 x – 8 ≥ 0

Problems
22. Solve – x 3 + 5 x 2 – 8 x + 4 ≥ 0.

23. Solve 3 x 2 ( x 2 – 8) + 6 x + 5 < 4 x 4 – 6 x (4 x – 1) + 4 algebraically. Display your result on a number line.

24. Create a cubic polynomial inequality for which x = 3 or x ≤ – 2 is the solution. Explain your reasoning.

Answer Section
MULTIPLE CHOICE
1. A 2. C
5. D 6. D
9. A 10. C
13. D 14. C
3. D 4. D
7. C 8. A
11. D 12. B
15. D 16. C
COMPLETION
17. family 18. inequality 19. 𝑥 ∈ ℝ - all the real numbers
SHORT ANSWER
20. a) x > 1.9 b) x ≤ – 13
21. a) x < –2, x > 1 b) −1 ≤ 𝑥 ≤ 0 , 1 ≤ 𝑥 ≤ 2 c) −3 ≤ 𝑥 ≤ 0, 𝑥 ≥ 3 d) 𝑥 ≤ −2, 𝑥 = 2
PROBLEM
22. Factor − x 3 + 5 x 2 – 8 x + 4 using the factor theorem.
− x 3 + 5 x 2 – 8 x + 4 = –( x – 2) 2 ( x – 1)
The inequality becomes –( x – 2) 2 ( x – 1) 0.
The solution is x ≤ 1 or x = 2.

23. 3 x 2 ( x 2 – 8) + 6 x + 5 < 4 x 4 – 6 x (4 x – 1) + 4
3 x 4 – 24 x 2 + 6 x + 5 < 4 x 4 – 24 x 2 + 6 x + 4 x 4 – 1 > 0
Factor x 4 – 1 as the difference of squares: x 4 – 1 = ( x – 1)( x + 1)( x 2 + 1).
The inequality becomes ( x – 1)( x + 1)( x 2 + 1) > 0.
Since x 2 + 1> 0 for all 𝑥 ∈ ℝ , it is equivalent to
( x – 1)( x + 1) > 0.
The solution is x < –1 or x > 1.
24. Answers may vary. Example: ( x + 2)( x – 3) 2 ≤ 0 .