To solve the polynomial division [tex]\((n^2 + 10n + 18) \div (n + 5)\)[/tex], follow these steps:
1. Set up the division:
We want to divide [tex]\(n^2 + 10n + 18\)[/tex] by [tex]\(n + 5\)[/tex]. This setup looks similar to long division.
2. Divide the leading term:
The leading term of the numerator is [tex]\(n^2\)[/tex], and the leading term of the denominator is [tex]\(n\)[/tex]. Divide these to get [tex]\(n: \)[/tex]
[tex]\[
n^2 \div n = n
\][/tex]
3. Multiply and subtract:
Multiply the entire divisor [tex]\((n + 5)\)[/tex] by the result from step 2 [tex]\((n)\)[/tex]:
[tex]\[
n(n + 5) = n^2 + 5n
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(n^2 + 10n + 18) - (n^2 + 5n) = 5n + 18
\][/tex]
4. Repeat the process:
Next, divide the new leading term [tex]\(5n\)[/tex] by [tex]\(n\)[/tex], yielding [tex]\(5\)[/tex]:
[tex]\[
5n \div n = 5
\][/tex]
Multiply the divisor [tex]\((n + 5)\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
5(n + 5) = 5n + 25
\][/tex]
Subtract this result from the remainder we got in the previous subtraction:
[tex]\[
(5n + 18) - (5n + 25) = 18 - 25 = -7
\][/tex]
5. Combine the results:
Now, putting it all together, we have the initial quotient components:
[tex]\[
n + 5
\][/tex]
The final quotient is [tex]\(n + 5\)[/tex] and the remainder is [tex]\(-7\)[/tex].
Therefore, the final result of dividing [tex]\(n^2 + 10n + 18\)[/tex] by [tex]\(n + 5\)[/tex] is:
[tex]\[
\boxed{n + 5 \quad \text{with a remainder of} \quad -7}
\][/tex]
