To perform the polynomial division of [tex]\( \left(n^3 - n^2 - 66n - 77\right) \)[/tex] by [tex]\( (n + 7) \)[/tex], we proceed as follows:
1. Setup the division:
[tex]\[
\frac{n^3 - n^2 - 66n - 77}{n + 7}
\][/tex]
2. Divide the first term of the numerator by the first term of the divisor:
- Divide [tex]\(n^3\)[/tex] by [tex]\(n\)[/tex], which gives [tex]\(n^2\)[/tex].
3. Multiply the entire divisor by this term:
- Multiply [tex]\(n^2\)[/tex] by [tex]\((n + 7)\)[/tex], resulting in [tex]\(n^3 + 7n^2\)[/tex].
4. Subtract this product from the original polynomial:
[tex]\[
n^3 - n^2 - 66n - 77 - (n^3 + 7n^2) = -8n^2 - 66n - 77
\][/tex]
5. Repeat the process with the new polynomial:
- Divide [tex]\(-8n^2\)[/tex] by [tex]\(n\)[/tex], which gives [tex]\(-8n\)[/tex].
- Multiply [tex]\(-8n\)[/tex] by [tex]\((n + 7)\)[/tex], resulting in [tex]\(-8n^2 - 56n\)[/tex].
- Subtract this product from the current polynomial:
[tex]\[
-8n^2 - 66n - 77 - (-8n^2 - 56n) = -10n - 77
\][/tex]
6. Continue with the resulting polynomial:
- Divide [tex]\(-10n\)[/tex] by [tex]\(n\)[/tex], which gives [tex]\(-10\)[/tex].
- Multiply [tex]\(-10\)[/tex] by [tex]\((n + 7)\)[/tex], resulting in [tex]\(-10n - 70\)[/tex].
- Subtract this product from the current polynomial:
[tex]\[
-10n - 77 - (-10n - 70) = -7
\][/tex]
7. Collect the quotient and the remainder:
- The quotient is [tex]\(n^2 - 8n - 10\)[/tex].
- The remainder is [tex]\(-7\)[/tex].
Therefore, the result of the division is:
[tex]\[
\frac{n^3 - n^2 - 66n - 77}{n + 7} = n^2 - 8n - 10 \quad \text{with a remainder of} \quad -7.
\][/tex]
So, the quotient is [tex]\(n^2 - 8n - 10\)[/tex] and the remainder is [tex]\(-7\)[/tex].
