Question

what is the remainder of the quotient x^4+4x^3-8x^2-8x+25 divided by x+5

Answer 1

To find the remainder of the polynomial \( p(x) = x^4 + 4x^3 - 8x^2 - 8x + 25 \) when divided by \( x + 5 \), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of the division of a polynomial \( p(x) \) by \( x - c \) is \( p(c) \).

In this case, we are dividing by \( x + 5 \), or equivalently \( x - (-5) \). Thus, we need to evaluate \( p(-5) \).

Calculating \( p(-5) \):

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

Calculating each term:

- \( (-5)^4 = 625 \)
- \( 4(-5)^3 = 4 \cdot (-125) = -500 \)
- \( -8(-5)^2 = -8 \cdot 25 = -200 \)
- \( -8(-5) = 40 \)

Now, substituting these values into the equation:

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

Now, simplify:

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

Thus, the remainder when \( p(x) \) is divided by \( x + 5 \) is \( \boxed{-10} \).