### A) Polynomial Long Division
First, let's set up the division:
[tex]\[
\frac{x^4 + x^3 + 7x^2 - 6x + 8}{x^2 + 2x + 8}
\][/tex]
1. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[
\frac{x^4}{x^2} = x^2
\][/tex]
Therefore, the first term of the quotient is [tex]\(x^2\)[/tex].
2. Multiply [tex]\(x^2\)[/tex] by the entire denominator:
[tex]\[
x^2 \cdot (x^2 + 2x + 8) = x^4 + 2x^3 + 8x^2
\][/tex]
3. Subtract this product from the original numerator:
[tex]\[
(x^4 + x^3 + 7x^2 - 6x + 8) - (x^4 + 2x^3 + 8x^2) = x^3 - x^2 - 6x + 8
\][/tex]
4. Divide the leading term of the new polynomial by the leading term of the denominator:
[tex]\[
\frac{x^3}{x^2} = x
\][/tex]
Therefore, the next term of the quotient is [tex]\(x\)[/tex].
5. Multiply [tex]\(x\)[/tex] by the entire denominator:
[tex]\[
x \cdot (x^2 + 2x + 8) = x^3 + 2x^2 + 8x
\][/tex]
6. Subtract this product from the new polynomial:
[tex]\[
(x^3 - x^2 - 6x + 8) - (x^3 + 2x^2 + 8x) = -3x^2 - 14x + 8
\][/tex]
7. Divide the leading term of the new polynomial by the leading term of the denominator:
[tex]\[
\frac{-3x^2}{x^2} = -1
\][/tex]
Therefore, the next term of the quotient is [tex]\(-1\)[/tex].
8. Multiply [tex]\(-1\)[/tex] by the entire denominator:
[tex]\[
-1 \cdot (x^2 + 2x + 8) = -x^2 - 2x - 8
\][/tex]
9. Subtract this product from the new polynomial:
[tex]\[
(-3x^2 - 14x + 8) - (-x^2 - 2x - 8) = -2x^2 - 12x + 16
\][/tex]
After this subtraction, we notice that it leads to zero, indicating no more terms can be divided. Therefore, our remainder is 0.
Hence, the quotient is:
[tex]\[
x^2 - x + 1
\][/tex]
The remainder is:
[tex]\[
0
\][/tex]
### B) Proof of the Answer
To verify the solution, we will multiply the quotient by the denominator and add the remainder. If our division is correct, this should reproduce the original numerator.
1. Quotient: [tex]\( x^2 - x + 1 \)[/tex]
2. Denominator: [tex]\( x^2 + 2x + 8 \)[/tex]
3. Remainder: [tex]\( 0 \)[/tex]
Multiply the quotient by the denominator:
[tex]\[
(x^2 - x + 1) \cdot (x^2 + 2x + 8)
\][/tex]
Distribute each term in the first polynomial to every term in the second polynomial:
[tex]\[
= x^2 \cdot x^2 + x^2 \cdot 2x + x^2 \cdot 8
= x^4 + 2x^3 + 8x^2
\][/tex]
[tex]\[
- x \cdot x^2 - x \cdot 2x - x \cdot 8
= -x^3 - 2x^2 - 8x
\][/tex]
[tex]\[
+ 1 \cdot x^2 + 1 \cdot 2x + 1 \cdot 8
= x^2 + 2x + 8
\][/tex]
Now, sum these products:
[tex]\[
x^4 + 2x^3 + 8x^2 - x^3 - 2x^2 - 8x + x^2 + 2x + 8
\][/tex]
Combine like terms:
[tex]\[
x^4 + (2x^3 - x^3) + (8x^2 - 2x^2 + x^2) + (-8x + 2x) + 8
= x^4 + x^3 + 7x^2 - 6x + 8
\][/tex]
As shown, multiplying the quotient by the denominator and adding the remainder yields exactly the original polynomial:
[tex]\[
x^4 + x^3 + 7x^2 - 6x + 8
\][/tex]
Therefore, our division was correct:
[tex]\[
\frac{x^4+x^3+7 x^2-6 x+8}{x^2+2 x+8} = x^2 - x + 1 \text{ with remainder } 0.
\][/tex]
