Question

What is the remainder of the quotient x4+4x3−8x2−8x+25÷x+5 ?(1 point) Responses −5 negative 5 1 1 −10 negative 10 7 7

Answer 1

To find the remainder of the polynomial \( x^4 + 4x^3 - 8x^2 - 8x + 25 \) when divided by \( x + 5 \), we can use the Remainder Theorem. The Remainder Theorem states that the remainder of \( f(x) \) when divided by \( x - c \) is \( f(c) \).

Here, we need to find \( f(-5) \):

\[
f(x) = x^4 + 4x^3 - 8x^2 - 8x + 25
\]

Now, substitute \( x = -5 \):

\[
f(-5) = (-5)^4 + 4(-5)^3 - 8(-5)^2 - 8(-5) + 25
\]

Calculating each term:

1. \( (-5)^4 = 625 \)
2. \( 4(-5)^3 = 4 \times (-125) = -500 \)
3. \( -8(-5)^2 = -8 \times 25 = -200 \)
4. \( -8(-5) = 40 \)
5. The constant term is \( 25 \).

Combining all these:

\[
f(-5) = 625 - 500 - 200 + 40 + 25
\]

Calculating step by step:

\[
625 - 500 = 125
\]
\[
125 - 200 = -75
\]
\[
-75 + 40 = -35
\]
\[
-35 + 25 = -10
\]

Thus, \( f(-5) = -10 \), which means the remainder when \( x^4 + 4x^3 - 8x^2 - 8x + 25 \) is divided by \( x + 5 \) is:

\[
\boxed{-10}
\]