MATH 121
THEORY OF POLYNOMIALS

IV. A. Write the polynomial of least degree with the given set of zeros and given leading coefficient.

1. {1 , 2 , -1 , i} ; 1

2. {2 + i , 2i} ; 2

3. {5 of multiplicity 2 , -3i} ; -3

4. {-i , 2 - i , i} ; 1

B. (5 - 8) Write the polynomial of least degree with real coefficients and the same zeros and leading coefficients as in A.

C. Find all zeros of the following polynomials. In problem 9, one zero is given.

9. P(x) = 2x ³ - 11x ² + 28x - 24 , r =

10. P(x) = x ² + 9

11. P(x) = 2x ² + 8

12. P(x) = x 4 - 1

13. P(x) = x ³ - 3x ² - 6x - 20

14. P(x) = x 4 – 2x 3 + 6x 2 + 22x + 13

D. (15 - 20) In (9 - 14) write each polynomial as the product of linear factors.

E. (21 - 26) In (9 - 14) write each polynomial as the product of linear or quadratic factors with real coefficients.

27. Find a polynomial function, f(x), of degree 3 with zeros: -2, 1, 5 and a y-intercept of 4.

ANSWERS

1)   (x - 1)(x + 1)(x - 2)(x - i)

2)   2(x - 2 - i)(x - 2i)

3)   -3(x - 5)2(x + 3i)

4) (x + i)(x - 2 + i)(x - i)

5)   (x - 1)(x + 1)(x - 2)(x - i)(x + i)

6)   2(x - 2 - i) (x - 2 + i )(x - 2i) (x + 2i)

7)   -3(x - 5)2(x - 3i) (x + 3i)

8)   (x - i)(x + i)(x - 2 - i)(x - 2 + i)(x - i)(x + i)

9)   , 2 ± 2i

10)   ±3i

11)   ±2i

12)   ±1 , ±i

13)   -1 ± i, 5

14)   -1, -1, 2 ± 3i

15)   P(x) = 2(x - )(x - 2 - 2i)(x - 2 + 2i)

16)   P(x) = (x - 3i) (x + 3i)

17)   P(x) = 2(x - 2i) (x + 2i)

18)   P(x) = (x - 1)(x + 1)(x - i)(x + i)

19)   P(x) = (x - 5)(x + 1 - i)(x + 1 + i)

20)   P(x)= (x + 1)2(x - 2 - 3i) (x - 2 + 3i)
21)   P(x) = 2(x - )(x2 - 4x + 8) 22)   P(x) = x2 + 9 23)   P(x) = 2(x ² + 4)
24)   P(x) = (x - 1)(x + 1)(x2 + 1) 25)   P(x) = (x - 5)(x2 + 2x + 4) 26)   P(x) = (x + 1)2(x2 - 4x + 13)
27)   f(x) = (2/5)(x + 2)(x - 1)(x - 5)