Sure, let's divide the polynomial [tex]\(2x^2 + 17x + 36\)[/tex] by [tex]\(x + 5\)[/tex]. We'll use the polynomial long division method.
### Step-by-Step Division:
1. Setup the division:
Write the dividend [tex]\(2x^2 + 17x + 36\)[/tex] and the divisor [tex]\(x + 5\)[/tex].
2. Divide the leading terms:
Divide the leading term of the dividend [tex]\(2x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex].
[tex]\[
\frac{2x^2}{x} = 2x
\][/tex]
So, the first term of the quotient is [tex]\(2x\)[/tex].
3. Multiply and subtract:
Multiply the entire divisor [tex]\(x + 5\)[/tex] by the first term of the quotient [tex]\(2x\)[/tex]:
[tex]\[
(2x)(x + 5) = 2x^2 + 10x
\][/tex]
Now, subtract this result from the original dividend:
[tex]\[
(2x^2 + 17x + 36) - (2x^2 + 10x) = 7x + 36
\][/tex]
4. Repeat the process:
Now, take the new dividend [tex]\(7x + 36\)[/tex] and divide its leading term [tex]\(7x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{7x}{x} = 7
\][/tex]
So, the next term of the quotient is [tex]\(7\)[/tex].
5. Multiply and subtract:
Multiply the entire divisor [tex]\(x + 5\)[/tex] by the new term of the quotient [tex]\(7\)[/tex]:
[tex]\[
(7)(x + 5) = 7x + 35
\][/tex]
Subtract this from the new dividend:
[tex]\[
(7x + 36) - (7x + 35) = 1
\][/tex]
### Results:
After completing the division, the quotient is [tex]\(2x + 7\)[/tex] and the remainder is [tex]\(1\)[/tex].
Thus, the division of [tex]\(2x^2 + 17x + 36\)[/tex] by [tex]\(x + 5\)[/tex] yields:
- Quotient: [tex]\(2x + 7\)[/tex]
- Remainder: [tex]\(1\)[/tex]
So, the final answer is:
[tex]\[
\boxed{(2x + 7, 1)}
\][/tex]
