Test 2-1 Outline
1. Use Long Division to divide f(x) by d(x) and write an answer in polynomial form.
f(x) = 5x4 + 14x3 + 9x ;d(x) = x2 + 3x
2. Use long division to find the quotient and remainder when dividing [pic] by 2x-1
3. Use the factor theorem to determine which of the following are factors of [pic]
x+2, x-3, x+4, x-5
4. Use the remainder theorem to find the remainder when f(x) is divided by x – k. Check your answer by using synthetic division.
f(x) = 3x3 - 2x2 + x – 5 ;k = -2
5. Use the rational zero theorem to come up with a list of potential rational zeros. Then determine which ones, if any are zeros.
f(x) = 2x4 - x3 - 4x2 - x – 6
6. Find all of the real zeros of the function, finding exact values whenever possible. Identify each zero as rational or irrational.
f(x) = x4- x3- 7x2 + 5x + 10
7. Use long division to find the quotient and remainder when
[pic]
8. Use synthetic division to find the quotient and remainder when
[pic]
9. Find the remainder of [pic] without doing long or synthetic division.
10. Use the factor theorem to determine which of the following are factors of [pic]. Work must be shown to justify your answer
a)[pic]
b)[pic]
c) [pic]
11. Find the remaining zeros of x3 – 8x2 + 9x + 6 if [pic] is itself a zero.
In problems 12-16 perform the indicated operation (without using a calculator) and write the result in a + bi form.
12. [pic]
13. [pic]
14. [pic]
15. [pic]
16. [pic]
17. Solve for x and y to make the equation true.
(3 – 2i) – 8 = x – (-7 + yi)
18. Simplify [pic]
19. Simplify [pic] and then [pic]
20. The graph and some additional information about a polynomial P(x) are presented below.
| • P(x) is a degree 5 polynomial with real coefficients. |[pic] |
|• The graph of P(x) for real numbers x is given on the grid. The only| |
|x-intercepts of P are at x = 4 and x = –3. | |
|• In the complex numbers, P(i+1) = 0. | |
|Find the polynomial P(x) | |
|Your final answer should be expressed as linear and irreducible | |
|quadratic factors with real coefficients. | |
| | |
21. You are given this information about a polynomial Q(x):
1. Q(x) is a degree 5 polynomial with real coefficients.
2. The graph of Q(x) for real numbers x is given on the grid. Its only x-intercepts are at x = –2 and x = 0.
3. Q(–1) = 4.
4. In the complex number system, Q(1 – 2i) = 0.
Find the factorization of Q(x) in the real number system.
22. Write a polynomial function in standard form (meaning it has real coefficients) whose degree is 3 and has zeros at 2 and 1 + i.
Practice Problems Solutions
1. Quotient: [pic], remainder: [pic]
2. Quotient: [pic], remainder: [pic]
3. x-5
4. -39
5. [pic]
6. x = -1 rational
x = 2 rational
x = [pic] irrational
7. Quotient: [pic], remainder: 0
8. Quotient: [pic], remainder: -520
9. -49
10. x -3
11. [pic]
12. 9 + 10i
13. -10 + 11i
14. [pic]
15. 0
16. -1-i
17. 34
18. 169
19. x = -12 and y = 2
20. -i
21. 2, and -6
22. [pic]
23. [pic]
24a. [pic]
24b. -27
25. [pic]
26. [pic]
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