Answer :
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]
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]
