Question

Use the Remainder Theorem to find the remainder when f(x) is divided by x-c. Then use the Factor Theorem to determine whether x-c is a factor of f(x). A. f(x) = 4x³ - 3x² - 8x + 4; c = 2 B. f(x) = -4x³ + 5x² + 8; c = -3 C. f(x) = 4x³ - 3x² - 8x + 4; c = -2 D. f(x) = -4x³ + 5x² + 8; c = 3

Answer 1

A) The remainder is 8.

B) The remainder is 161.

C) The remainder is -24.

D) The remainder is -55.

Step-by-step explanation:

The Remainder Theorem states that if a polynomial f(x) is divided by x - c, the remainder is f(c). To find the remainder for each given function and value of c, we substitute c into the polynomial.

• A. f(2) = 4(2)³ - 3(2)² - 8(2) + 4 = 32 - 12 - 16 + 4 = 8. Thus, the remainder is 8.
• B. f(-3) = -4(-3)³ + 5(-3)² + 8 = -4(-27) + 5(9) + 8 = 108 + 45 + 8 = 161. The remainder is 161.
• C. f(-2) = 4(-2)³ - 3(-2)² - 8(-2) + 4 = -32 - 12 + 16 + 4 = -24. The remainder is -24.
• D. f(3) = -4(3)³ + 5(3)² + 8 = -4(27) + 5(9) + 8 = -108 + 45 + 8 = -55. The remainder is -55.

Next, the Factor Theorem states that x - c is a factor of f(x) if and only if f(c) = 0. From the calculations, none of the remainders are zero, which means that in all cases, x - c is not a factor of f(x).

Therefore,

A) The remainder is 8.

B) The remainder is 161.

C) The remainder is -24.

D) The remainder is -55.