Certainly! Let's solve the problem step-by-step.
We are given the polynomial [tex]\( P(x) = x^4 - x^3 - 7x^2 + x + k \)[/tex] and the divisor [tex]\( D(x) = x + 2 \)[/tex]. We need to find the value of [tex]\( k \)[/tex] such that when [tex]\( P(x) \)[/tex] is divided by [tex]\( D(x) \)[/tex], the remainder is 10.
1. Step 1: Perform Polynomial Division
We start by dividing [tex]\( P(x) \)[/tex] by [tex]\( D(x) \)[/tex]. Using polynomial long division (or synthetic division), we divide [tex]\( x^4 - x^3 - 7x^2 + x + k \)[/tex] by [tex]\( x + 2 \)[/tex].
The quotient is [tex]\( Q(x) = x^3 - 3x^2 - x + 3 \)[/tex] and the remainder [tex]\( R \)[/tex].
Therefore, we can express [tex]\( P(x) \)[/tex] as:
[tex]\[
P(x) = (x + 2)(x^3 - 3x^2 - x + 3) + R
\][/tex]
2. Step 2: Find the Remainder [tex]\( R \)[/tex]
Given the quotient [tex]\( Q(x) \)[/tex], when substituting [tex]\( x + 2 \)[/tex]:
[tex]\[
P(x) = x^4 - x^3 - 7x^2 + x + k = (x + 2)(x^3 - 3x^2 - x + 3) + R
\][/tex]
By substituting [tex]\( x = -2 \)[/tex] into the equation, the terms involving [tex]\( x \)[/tex] will cancel out:
[tex]\[
P(-2) = (-2)^4 - (-2)^3 - 7(-2)^2 + (-2) + k
\][/tex]
[tex]\[
P(-2) = 16 + 8 - 28 - 2 + k
\][/tex]
Simplifying this, we get:
[tex]\[
P(-2) = 16 + 8 - 28 - 2 + k = -6 + k
\][/tex]
3. Step 3: Set the Remainder Equal to 10
Since we need the remainder to be 10, we set up the equation:
[tex]\[
-6 + k = 10
\][/tex]
4. Step 4: Solve for [tex]\( k \)[/tex]
Solving for [tex]\( k \)[/tex] gives:
[tex]\[
k = 16
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] that makes the remainder 10 when [tex]\( x^4 - x^3 - 7x^2 + x + k \)[/tex] is divided by [tex]\( x + 2 \)[/tex] is [tex]\( \boxed{16} \)[/tex].
