MATH 121
THEORY OF POLYNOMIALS

III. A. Use the remainder theorem to find:

1. P(1) if P(x) = x² - x + 3

2. P(0) if P(x) = 3x³ + x² - 5

3. P(2) if P(x) = 3x4 – x2 + 15

    B. Show which of the following linear expressions is a factor of the given polynomial:

4. Is x - 3 a factor of  3x³ + 2x² + x - 1?

5. Is x + 2 a factor of  x4 – x3 –2x + 3?

6. Is x - 3 a factor of 2x³ - 9x² + 14x - 15?

7. Is x + 2 a factor of x4 + 5x3 – 5x2 – 3x – 2?

    C. Use the factor theorem to show:

8. x - 2 is a factor of x³ - 5x² - 3x + 18

9. x - 1 is a factor of x³ - 2x² - 2x + 3

10. x + 3 is a factor of 2x³ + 5x² - 4x - 3

11. x + 1 is a factor of x³ + x² + x + 1

12. x - i is a factor of x³ + x² + x + 1

    D. Find the remainder if:

13. x10 + x3 + i is divided by x + 1

14. x³ + x is divided by x + 10

15. x342 + 13 is divided by x - i

16. x342 + 13 is divided by x + i

    E.  17. For what values of K does x² + Kx + 2 give the same remainder when divided by (x - 1) or (x + 1)?

18. If P(x) = x5 + Ax3 + Ax + 4 is such that P(2) = 6, find P(-2).

19. When x² + x - 3 is divided by x - r, the remainder is -1.   Find r.

ANSWERS

1) 3

2) -5

3) 59

4) no

5) no

6) yes

7) yes

8) P(2) = 0

9) P(1) = 0

10) P(-3) = 0

11) P(-1) = 0

12) P(i) = 0

13) i

14) -1010

15) 12

16) 12

17) K = 0

18) 2

19) 1 , -2