Question

What is the remainder of the quotient x^4+4x^3−8x^2−8x+25÷x+5 ?(1 point)

Answer 1

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

In this case, we can express the divisor \( x + 5 \) as \( x - (-5) \). Thus, we need to find \( f(-5) \) where \( f(x) = x^4 + 4x^3 - 8x^2 - 8x + 25 \).

First, we will evaluate \( f(-5) \):

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

Calculating each term step by step:

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

Now, combine all these results:

\[
f(-5) = 625 - 500 - 200 + 40 + 25
\]
\[
= 625 - 500 = 125
\]
\[
= 125 - 200 = -75
\]
\[
= -75 + 40 = -35
\]
\[
= -35 + 25 = -10
\]

Thus, the remainder of the quotient when \( x^4 + 4x^3 - 8x^2 - 8x + 25 \) is divided by \( x + 5 \) is:

\[
\boxed{-10}
\]