MATH 121
THEORY OF POLYNOMIALS

II. A. Use synthetic division to find the quotient, Q(x) and the remainder R if P(x) is the dividend and D(x) is the divisor.

1. P(x) = 3x 4 - 5x3 - 3x2 + 11x - 10 2. P(x) = x 4 + 2x3 - 10x - 12 3. P(x) = 4x3 + 5x2 - 23x + 6 4. P(x) = x 4 + 5x3 + 7x ² - 8x + 9 5. P(x) = x + 3x3 - 7x 4 + 2 6. P(x) = x9 + 1 7. P(x) = 3x3 - ix2 + 2x - 3i

D(x) = x - 5 D(x) = x + 3 D(x) = x - 2 D(x) = x + 5 D(x) = x - 1 D(x) = x - 1 D(x) = x - i

B. Determine whether the linear expressions given are factors of the given polynomial P(x).

8. P(x) = x³ - 3x² + 3x - 1

a) x - 1     b) x + 1     c) x - 2

9. P(x) = x⁵ + 11x³ - 12x

a) x - 1     b) x - 3     c) x + 5

ANSWERS

1. Q(x) = 3x³ + 10x² + 47x + 246; R = 1220

2. Q(x) = x³ - x² + 3x - 19; R = 45

3. Q(x) = 4x² + 13x + 3; R = 12

4. Q(x) = x³ + 7x - 43; R = 224

5. Q(x) = -7x³ - 4x² - 4x - 3; R = -1

6. Q(x) = x⁸ + x⁷ + x⁶ + x ⁵ + x ⁴ + x³ + x + x + 1; R = 2

7. Q(x) = 3x² + 2ix; R = -3i

8. yes , no , no

9. yes, no, no