Sure, let's rewrite the given expression [tex]\(\frac{2x^2 + 13x + 26}{x+4}\)[/tex] in the form [tex]\(q(x) + \frac{r(x)}{b(x)}\)[/tex], where [tex]\(b(x)\)[/tex] is the original denominator.
### Step-by-Step Solution:
1. Set Up Polynomial Division: We need to divide the polynomial numerator [tex]\(2x^2 + 13x + 26\)[/tex] by the polynomial denominator [tex]\(x + 4\)[/tex]. The goal is to find the quotient [tex]\(q(x)\)[/tex] and the remainder [tex]\(r(x)\)[/tex].
2. Perform the Division:
- Find the first term of the quotient by dividing the leading term of the numerator [tex]\(2x^2\)[/tex] by the leading term of the denominator [tex]\(x\)[/tex], which gives [tex]\(2x\)[/tex].
- Multiply [tex]\(2x\)[/tex] by the denominator [tex]\(x + 4\)[/tex], resulting in [tex]\(2x^2 + 8x\)[/tex].
- Subtract this from the original numerator:
[tex]\[
(2x^2 + 13x + 26) - (2x^2 + 8x) = 5x + 26
\][/tex]
- Repeat the process with the new numerator [tex]\(5x + 26\)[/tex]. Divide [tex]\(5x\)[/tex] by [tex]\(x\)[/tex], yielding [tex]\(5\)[/tex].
- Multiply [tex]\(5\)[/tex] by the denominator [tex]\(x + 4\)[/tex] to get [tex]\(5x + 20\)[/tex].
- Subtract this from the current numerator:
[tex]\[
(5x + 26) - (5x + 20) = 6
\][/tex]
3. Identify Quotient and Remainder:
- The quotient [tex]\(q(x)\)[/tex] from the division is [tex]\(2x + 5\)[/tex].
- The remainder [tex]\(r(x)\)[/tex] is [tex]\(6\)[/tex].
4. Rewrite the Expression:
[tex]\[
\frac{2x^2 + 13x + 26}{x + 4} = (2x + 5) + \frac{6}{x + 4}
\][/tex]
Therefore, in the rewritten expression, [tex]\(q(x)\)[/tex] is [tex]\(2x + 5\)[/tex] and [tex]\(r(x)\)[/tex] is [tex]\(6\)[/tex].
