Math 3
Module 3 Test Review
Name ______________________________
Period: ________
1. What type of function will result when a linear function is added to a linear function?
Explain your reasoning.
2. What type of function will result when a linear function is multiplied by a linear function?
Explain your reasoning.
3. What type of function will result when a linear function is multiplied by a quadratic function?
Explain your reasoning.
4. a. How many distinct roots does the function 𝑓(𝑥) = (𝑥 + 1)(𝑥 − 3)(𝑥 + 4) have?
b. Find each root and state its’ multiplicity.
c. At each root, identify if the graph will cross or touch and turn the x-axis?
5. a. How many distinct roots does the function 𝑓(𝑥) = 𝑥(𝑥 + 7)(𝑥 + 2)3 (𝑥 − 34)5 have?
b. Find each root and state its’ multiplicity.
c. At each root, identify if the graph will cross or touch and turn the x-axis?
6. a. Is (𝑥 − 7) a factor of 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 − 24𝑥 − 28?
b. Factor 𝑓(𝑥) completely.
c. At each root, identify if the graph will cross or touch and turn the x-axis?
7. What is the function with the given roots of 2 and -5?
8. What is the function with a root at 3 with multiplicity of 2 and another root at -4?
9. What is the function with a root at -5 with multiplicity of 3, root 2 with multiplicity of 2 and a
root at 0 with multiplicity of 1? Write in factored form.
10. What is the end behavior of the function?
a. 𝑓(𝑥) = (𝑥 − 3)(𝑥 − 5)(2 − 𝑥)
b.
End Behavior
𝐴𝑠 𝑥 → −∞ 𝑓(𝑥) →________
𝐴𝑠 𝑥 → ∞ 𝑓(𝑥) →__________
𝑔(𝑥) = (𝑥 + 2)2 (𝑥 + 6)
End Behavior
𝐴𝑠 𝑥 → −∞ 𝑓(𝑥) →________
𝐴𝑠 𝑥 → ∞ 𝑓(𝑥) →__________
c. ℎ(𝑥) has a root at -3, another root at 5 with multiplicity of 2. Write the function h(x) and state
the end behavior.
Function:
End Behavior
𝐴𝑠 𝑥 → −∞ 𝑓(𝑥) →________
𝐴𝑠 𝑥 → ∞ 𝑓(𝑥) →__________
Use the given functions to perform the indicated operations.
𝑓(𝑥) = 𝑥 + 6
𝑔(𝑥) = 3𝑥 − 1
ℎ(𝑥) = 𝑥 2 + 4𝑥 − 5
11. 𝑓(𝑥) + 𝑔(𝑥)
12.
𝑔(𝑥) − 𝑓(𝑥)
14. 𝑓(𝑥) ∗ 𝑔(𝑥)
15.
ℎ(𝑥) ∗ 𝑔(𝑥)
13. 𝑔(𝑥) − ℎ(𝑥)
16. State the type of function that resulted from problems 11-15. (Linear, Quadratic, Cubic,
Exponential)
11: _____________
14: ___________
12:_____________
15: ___________
13: _____________
17. Find the x-intercepts and state the multiplicity of each factor, find the y-intercept and the end
behavior. Graph the Polynomial using this information.
𝑓(𝑥) = (𝑥 + 3)(𝑥 − 2)(𝑥 + 1)
x-intercept(s):
Multiplicity of each factor:
y-intercept:
End Behavior
𝐴𝑠 𝑥 → −∞ 𝑓(𝑥) →________
𝐴𝑠 𝑥 → ∞ 𝑓(𝑥) →________
18. Find the x-intercepts and state the multiplicity of each factor, find the y-intercept and the end
behavior. Graph the Polynomial using this information.
𝑓(𝑥) = (𝑥 − 1)(𝑥 2 − 6𝑥 + 9)
x-intercept(s):
Multiplicity of each factor:
y-intercept:
End Behavior
𝐴𝑠 𝑥 → −∞ 𝑓(𝑥) →________
𝐴𝑠 𝑥 → ∞ 𝑓(𝑥) →________
18. Find the x-intercepts and state the multiplicity of each factor, find the y-intercept and the end
behavior. Graph the Polynomial using this information.
𝑓(𝑥) = 𝑥 3 + 6𝑥 2 + 11𝑥 + 6 , one root is -1.
x-intercept(s):
Multiplicity of each factor:
y-intercept:
End Behavior
𝐴𝑠 𝑥 → ∞ 𝑓(𝑥) →________
𝐴𝑠 𝑥 → −∞ 𝑓(𝑥) →________
17. Find the missing information given the x-intercepts and graph the function.
Equation: 𝑓(𝑥) =
x-intercept(s): 2 with multiplicity of 1,
-3 with multiplicity of 1,
and 1 with multiplicity of 2
y-intercept:
End Behavior
𝐴𝑠 𝑥 → ∞ 𝑓(𝑥) →________
𝐴𝑠 𝑥 → −∞ 𝑓(𝑥) →________
18.
A polynomial with rational roots has the following zeros. Find the remaining zeros.
a.
3, 4 − √3
b.
2√5 , 0 , −6 + 3√11
c.
-5, 2, 2 + 3i
19.
Find all the roots of the polynomial if one is given. 𝑓(𝑥) = 2𝑥 3 + 3𝑥 2 − 65𝑥 + 84 ; x = 4
20.
Find all the factors of the polynomial if one is given. 𝑓(𝑥) = 𝑥 3 − 𝑥 2 − 11𝑥 + 3; (x+3)