Use the Remainder Theorem to find the remainder when the function f left parenthesis x right parenthesis equals x cubed plus 8 x squared minus 2 x is divided by left parenthesis x plus 3 right parenthesis.

Question

(1 point)
Responses

negative 93
Image with alt text: negative 93

93
93

51
51

39
39

Answer 1

The Remainder Theorem states that the remainder of the division of a polynomial \( f(x) \) by a linear divisor of the form \( x - c \) is \( f(c) \).

In this case, we need to find the remainder when dividing \( f(x) = x^3 + 8x^2 - 2x \) by \( x + 3 \). We first rewrite the divisor in the form \( x - c \) by letting \( c = -3 \).

Now, we will calculate \( f(-3) \):

\[ f(-3) = (-3)^3 + 8(-3)^2 - 2(-3) \]

Calculating each term:

• \( (-3)^3 = -27 \)
• \( 8(-3)^2 = 8 \cdot 9 = 72 \)
• \( -2(-3) = 6 \)

Now, putting it all together: \[ f(-3) = -27 + 72 + 6 \] \[ = -27 + 72 = 45 \] \[ = 45 + 6 = 51 \]

Therefore, the remainder when \( f(x) \) is divided by \( x + 3 \) is 51.

The answer is 51.