Question
What is the remainder to the following polynomial division problem?
(x^2 +6x +12) / (x+1)
(x^2 +6x +12) / (x+1)
Answers
GPT-4o mini
To find the remainder of the polynomial division of \( (x^2 + 6x + 12) \) by \( (x + 1) \), we can use the Remainder Theorem, which states that the remainder of the polynomial \( f(x) \) when divided by \( x - c \) is \( f(c) \).
In this case, we need to evaluate \( f(-1) \) since we are dividing by \( x + 1 \) (or \( x - (-1) \)).
1. First, identify the polynomial:
\[
f(x) = x^2 + 6x + 12
\]
2. Now, substitute \( x = -1 \):
\[
f(-1) = (-1)^2 + 6(-1) + 12
\]
\[
= 1 - 6 + 12
\]
\[
= 1 + 6
\]
\[
= 7
\]
Thus, the remainder when dividing \( (x^2 + 6x + 12) \) by \( (x + 1) \) is \( \boxed{7} \).
In this case, we need to evaluate \( f(-1) \) since we are dividing by \( x + 1 \) (or \( x - (-1) \)).
1. First, identify the polynomial:
\[
f(x) = x^2 + 6x + 12
\]
2. Now, substitute \( x = -1 \):
\[
f(-1) = (-1)^2 + 6(-1) + 12
\]
\[
= 1 - 6 + 12
\]
\[
= 1 + 6
\]
\[
= 7
\]
Thus, the remainder when dividing \( (x^2 + 6x + 12) \) by \( (x + 1) \) is \( \boxed{7} \).