Answer/Step-by-step explanation:
Based on the factor theorem, a polynomial, x - a, is said to be a factor of another polynomial, f(x), if an only if f(a) = 0.
Using this theorem, let's determine whether each of the given first polynomial, is a factor of the second polynomial.
1. x - 1; x² + 2x + 5
By the factor theorem, x - 1 will be a factor of f(x) = x² + 2x + 5, if and only if f(1) = 0.
f(1) = 1² + 2(1) + 5
= 1 + 2 + 5
= 8
Since f(1) ≠ 0, therefore the first polynomial, x - 1, is NOT a factor of the second polynomial, x² + 2x + 5.
2. x + 1; x³ - x - 2
By the factor theorem, x + 1 will be a factor of f(x) = x³ - x - 2, if and only if f(-1) = 0.
f(1) = -1³ - (-1) - 2
= -1 + 1 - 2
= -2
Since f(-1) ≠ 0, therefore the first polynomial, x + 1, is NOT a factor of the second polynomial, x³ - x - 2.
3. x - 4; 2x³ - 9x² + 9x - 20
By the factor theorem, x - 4 will be a factor of f(x) = 2x³ - 9x² + 9x - 20, if and only if f(4) = 0.
f(4) = 2(4)³ - 9(4)² + 9(4) - 20
= 128 - 144 + 36 - 20
= 0
Since f(4) = 0, therefore the first polynomial, x + 4, is a factor of the second polynomial, 2x³ - 9x² + 9x - 20.
4. a - 1; a³ - 2a² + a - 2
By the factor theorem, a - 1 will be a factor of f(a) = a³ - 2a² + a - 2, if and only if f(1) = 0.
f(1) = 1³ - 2(1)² + 1 - 2
= 1 - 2 + 1 - 2
= -2
Since f(1) ≠ 0, therefore the first polynomial, a - 1, is NOT a factor of the second polynomial, a³ - 2a² + a - 2.
5. y + 3; 2y³ + y² - 13y + 6
By the factor theorem, y + 3 will be a factor of f(y) = 2y³ + y² - 13y + 6, if and only if f(-3) = 0.
f(-3) = 2(-3)³ + (-3)² - 13(-3) + 6
= -54 + 9 + 39 + 6
= 0
Since f(-3) = 0, therefore the first polynomial, y + 3, is a factor of the second polynomial, 2y³ + y² - 13y + 6.
