Answer: a) Factored form: (x - 5)(x² + 7x - 7) ; P(5) = 0 Yes
b) Factored form: (x + 1)(x³- 5x² - 4x - 1) ; P(-1) = 0 Yes
Step-by-step explanation:
Use synthetic division to verify the remainder (last digit) is zero. The other digits represent the coefficients of the reduced polynomial.
a) x³ + 2x² - 42x + 35 ; x - 5 = 0 --> x = 5
5 | 1 2 -42 35
| ↓ 5 35 -35
1 7 -7 0 → remainder is zero
Reduced polynomial is: x² + 7x - 7
Factored form: P(x) = (x - 5)(x² + 7x - 7)
b) x⁴ - 4x³ -9x² - 5x - 1 ; x + 1 = 0 --> x = -1
-1 | 1 -4 -9 -5 -1
| ↓ -1 5 4 1
1 -5 -4 -1 0 → remainder is zero
Reduced polynomial is: x³- 5x² - 4x - 1
Factored form: P(x) = (x + 1)(x³- 5x² - 4x - 1)
