Advanced Math
Quiz 3.1-3.3 Review
Name:
Dec. 2012
Use Synthetic Division to divide the first polynomial by the second polynomial.
1. 5𝑥 3 + 6𝑥 2 − 8 𝑥 + 1, 𝑥 − 5
1.
Quotient:__________________
Remainder:________________
2. −𝑥 5 − 10𝑥 3 + 5 𝑥 − 1, 𝑥 + 4
2.
Quotient:__________________
Remainder:________________
3. 12𝑥 3 + 5𝑥 2 + 5 𝑥 − 7, 𝑥 +
3
4
3.
Quotient:__________________
Remainder:________________
4. 8𝑥 3 − 4𝑥 2 + 6 𝑥 − 3, 𝑥 −
1
2
4.
Quotient:__________________
Remainder:________________
5. 2𝑥 5 − 3𝑥 4 − 5𝑥 2 − 10, 𝑥 − 4
5.
Quotient:__________________
Remainder:________________
6. 𝑥 6 − 1, 𝑥 + 1
6.
Quotient:__________________
Remainder:________________
Use Synthetic Division and the Remainder Theorem to find 𝑷(𝒄).
7. 𝑃 (𝑥) = 2𝑥 3 − 𝑥 2 + 3𝑥 − 1 , 𝑐 = 3
7.
Remainder with Synthetic Division:_________________
Remainder with the Remainder Theorem:____________
8. 𝑃 (𝑥) = 6𝑥 3 − 𝑥 2 + 4𝑥 , 𝑐 = −3
8.
Remainder with Synthetic Division:_________________
Remainder with the Remainder Theorem:____________
9. 𝑃 (𝑥) = 𝑥 5 + 20𝑥 2 − 1 , 𝑐 = −4
9.
Remainder with Synthetic Division:_________________
Remainder with the Remainder Theorem:____________
10. 𝑃 (𝑥) = −𝑥 3 + 3𝑥 2 + 5𝑥 + 30 , 𝑐 = 8
10.
Remainder with Synthetic Division:_________________
Remainder with the Remainder Theorem:____________
Use Synthetic Division and the Factor Theorem to determine whether the given binomial is a factor of 𝑷(𝒙).
11. 𝑃 (𝑥) = 𝑥 3 + 4𝑥 2 − 27𝑥 − 90 , 𝑥 + 6
11._______________________
12. 𝑃 (𝑥) = 3𝑥 3 + 4𝑥 2 − 27𝑥 − 36 , 𝑥 − 4
12._______________________
1
13. 𝑃 (𝑥) = 16𝑥 4 − 8𝑥 3 + 9𝑥 2 + 14𝑥 + 4 , 𝑥 − 4
13._______________________
14. 𝑃 (𝑥) = 𝑥 5 + 2𝑥 4 − 22𝑥 3 − 50𝑥 2 − 75𝑥 , 𝑥 − 5
14._______________________
Examine the leading term and the degree of the polynomial to determine the far-left and far-right behavior of the
graph.
15. 𝑓(𝑥) = 2𝑥 4 − 3𝑥 2 − 5𝑥 + 1
Degree___________
Sign of Leading Coefficient_________________
End Behavior:_______________________________________________
_____________________________________________
As 𝑥 → ∞, 𝑓(𝑥) → _________
16. 𝑓(𝑥) = −6𝑥 3 − 9𝑥 2 + 15𝑥 − 3
Degree___________
As 𝑥 → −∞, 𝑓(𝑥) → _________
Sign of Leading Coefficient_________________
End Behavior:_______________________________________________
_____________________________________________
As 𝑥 → ∞, 𝑓(𝑥) → _________
1
2
17. 𝑓(𝑥) = 𝑥 5 − 6𝑥 3 − 12𝑥 2 + 7
Degree___________
As 𝑥 → −∞, 𝑓(𝑥) → _________
Sign of Leading Coefficient_________________
End Behavior:_______________________________________________
_____________________________________________
As 𝑥 → ∞, 𝑓(𝑥) → _________
18. 𝑓(𝑥) = −3𝑥 4 + 4𝑥 3 − 5𝑥 2 − 𝑥 + 6
Degree___________
As 𝑥 → −∞, 𝑓(𝑥) → _________
Sign of Leading Coefficient_________________
End Behavior:_______________________________________________
_____________________________________________
As 𝑥 → ∞, 𝑓(𝑥) → _________
19. 𝑓(𝑥) = −4𝑥 + 4 − 𝑥 2
Degree___________
As 𝑥 → −∞, 𝑓(𝑥) → _________
Sign of Leading Coefficient_________________
End Behavior:_______________________________________________
_____________________________________________
As 𝑥 → ∞, 𝑓(𝑥) → _________
20. 𝑓(𝑥) = −81 + 𝑥 4
Degree___________
As 𝑥 → −∞, 𝑓(𝑥) → _________
Sign of Leading Coefficient_________________
End Behavior:_______________________________________________
_____________________________________________
As 𝑥 → ∞, 𝑓(𝑥) → _________
As 𝑥 → −∞, 𝑓(𝑥) → _________
Determine (by estimation) the maximum and minimum values .
21a.
21b.
21c.
Max:__________________
Max:__________________
Max:__________________
Min:__________________
Min:__________________
Min:__________________
For the graphs above, determine the “far-right” and “far-left” behavior.
22a. As 𝑥 → ∞, 𝑓(𝑥) →
As 𝑥 → −∞, 𝑓(𝑥) →
22b. As 𝑥 → ∞, 𝑓(𝑥) →
As 𝑥 → −∞, 𝑓(𝑥) →
22c. As 𝑥 → ∞, 𝑓(𝑥) →
As 𝑥 → −∞, 𝑓(𝑥) →
Find the real zeros of each polynomial function by factoring. The number in parentheses to the right of each
polynomial indicates the number of real zeros of the given polynomial function.
23. 𝑃(𝑥) = 𝑥 3 − 2𝑥 2 − 24𝑥
24. 𝑃(𝑥) = 𝑥 4 − 5𝑥 2 + 4
(3)
(4)
25. 𝑃(𝑥) = 𝑥 4 − 29𝑥 2 + 100
26. 𝑃(𝑥) = 𝑥 3 − 7𝑥 2 + 10𝑥
23._______________________
24._______________________
(4)
(3)
25._______________________
26._______________________
Use the Intermediate Value Theorem to verify that 𝑷(𝒙) has a zero between 𝒂 𝒂𝒏𝒅 𝒃. Explain why there is a zero
between a and b.
27. 𝑃(𝑥) = 4𝑥 3 − 𝑥 2 − 6𝑥 + 1; 𝑎 = 0, 𝑏 = 1
27.
_______________________________________
_______________________________________
28. 𝑃(𝑥) = 5𝑥 3 − 16𝑥 2 − 20𝑥 + 64; 𝑎 = 3, 𝑏 = 3.5
28.
_______________________________________
_______________________________________
29. 𝑃(𝑥) = 𝑥 3 − 𝑥 − 2; 𝑎 = 1.5, 𝑏 = 1.6
29.
_______________________________________
_______________________________________
30. 𝑃(𝑥) = −𝑥 3 − 2𝑥 2 + 𝑥 − 3; 𝑎 = −2.8, 𝑏 = −2.7
30.
_______________________________________
_______________________________________
Procedure for graphing:
1. Start by graphing the zeros
2. Then determine whether the graph passes through the zero or hits and bounces off the zero
3. Graph (if possible) the y-intercept
4. Determine the end behavior – which way should the arrows go?
5. Create a smooth curve
31. 𝑃(𝑥) = (𝑥 + 1)(𝑥 − 2)(𝑥 + 5)
What is the degree of the polynomial?_________
What kind of number is the degree?___________
What is the sign of the leading coefficient?_____
What is the end behavior?
As 𝑥 → ∞, 𝑓(𝑥) →
As 𝑥 → −∞, 𝑓(𝑥) →
What are the x-intercepts (zeros) & their multiplicities?
_____________ multiplicity_______ pass through/bounce
_____________ multiplicity_______ pass through/bounce
_____________ multiplicity_______ pass through/bounce
What is the y-intercept? [plug in x = 0] _______________
You will only graph this if it fits – if it doesn’t, just estimate!
32. 𝑃(𝑥) = (𝑥 − 4)2 (𝑥 + 1)
What is the degree of the polynomial?_________
What kind of number is the degree?___________
What is the sign of the leading coefficient?_____
What is the end behavior?
As 𝑥 → ∞, 𝑓(𝑥) →
As 𝑥 → −∞, 𝑓(𝑥) →
What are the x-intercepts (zeros) & their multiplicities?
_____________ multiplicity_______ pass through/bounce
_____________ multiplicity_______ pass through/bounce
What is the y-intercept? [plug in x = 0] _______________
You will only graph this if it fits – if it doesn’t, just estimate!
33. 𝑃(𝑥) = −(𝑥 − 2)2 (𝑥 + 5)
What is the degree of the polynomial?_________
What kind of number is the degree?___________
What is the sign of the leading coefficient?_____
What is the end behavior?
As 𝑥 → ∞, 𝑓(𝑥) →
As 𝑥 → −∞, 𝑓(𝑥) →
What are the x-intercepts (zeros) & their multiplicities?
_____________ multiplicity_______ pass through/bounce
_____________ multiplicity_______ pass through/bounce
What is the y-intercept? [plug in x = 0] _______________
You will only graph this if it fits – if it doesn’t, just estimate!
34. 𝑃(𝑥) = 𝑥(𝑥 − 2)2
What is the degree of the polynomial?_________
What kind of number is the degree?___________
What is the sign of the leading coefficient?_____
What is the end behavior?
As 𝑥 → ∞, 𝑓(𝑥) →
As 𝑥 → −∞, 𝑓(𝑥) →
What are the x-intercepts (zeros) & their multiplicities?
_____________ multiplicity_______ pass through/bounce
_____________ multiplicity_______ pass through/bounce
What is the y-intercept? [plug in x = 0] _______________
You will only graph this if it fits – if it doesn’t, just estimate!
35. 𝑃(𝑥) = (𝑥 − 1)2 (𝑥 + 3)2
What is the degree of the polynomial?_________
What kind of number is the degree?___________
What is the sign of the leading coefficient?_____
What is the end behavior?
As 𝑥 → ∞, 𝑓(𝑥) →
As 𝑥 → −∞, 𝑓(𝑥) →
What are the x-intercepts (zeros) & their multiplicities?
_____________ multiplicity_______ pass through/bounce
_____________ multiplicity_______ pass through/bounce
What is the y-intercept? [plug in x = 0] _______________
You will only graph this if it fits – if it doesn’t, just estimate!
36. 𝑃(𝑥) = −(𝑥 − 2)(𝑥 − 1)2 (𝑥 + 4)
What is the degree of the polynomial?_________
What kind of number is the degree?___________
What is the sign of the leading coefficient?_____
What is the end behavior?
As 𝑥 → ∞, 𝑓(𝑥) →
As 𝑥 → −∞, 𝑓(𝑥) →
What are the x-intercepts (zeros) & their multiplicities?
_____________ multiplicity_______ pass through/bounce
_____________ multiplicity_______ pass through/bounce
_____________ multiplicity_______ pass through/bounce
What is the y-intercept? [plug in x = 0] _______________
You will only graph this if it fits – if it doesn’t, just estimate!
37. 𝑃(𝑥) = −𝑥 2 (𝑥 + 3)2
What is the degree of the polynomial?_________
What kind of number is the degree?___________
What is the sign of the leading coefficient?_____
What is the end behavior?
As 𝑥 → ∞, 𝑓(𝑥) →
As 𝑥 → −∞, 𝑓(𝑥) →
What are the x-intercepts (zeros) & their multiplicities?
_____________ multiplicity_______ pass through/bounce
_____________ multiplicity_______ pass through/bounce
What is the y-intercept? [plug in x = 0] _______________
You will only graph this if it fits – if it doesn’t, just estimate!