QQuestionMathematics
QuestionMathematics
Instructions: Select the correct answer.
Rewrite the expression $\frac{x^{3}+ 10 x^{2}+ 13 x+ 39}{x^{2}+ 2 x+ 1}$ in the form $q(x)+\frac{r(x)}{b(x)}$.
\begin{aligned}
& (x+ 8)+\frac{- 4 x+ 31}{x^{2}+ 2 x+ 1} \\
& (- 4 x+ 31)+\frac{x+ 8}{x^{2}+ 2 x+ 1} \\
& (x+ 8)+\frac{- 4 x+ 31}{x^{3}+ 10 x^{2}+ 13 x+ 39} \\
& (- 4 x+ 31)+\frac{x+ 8}{x^{3}+ 10 x^{2}+ 13 x+ 39}
\end{aligned}
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Answer
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Step 1:I'll solve this step by step using polynomial long division.
Step 2:: Set up polynomial long division
We want to divide $$\frac{x^{3}+10x^{2}+13x+39}{x^{2}+2x+1}
Step 3:: Perform polynomial long division
- This leaves $$8x^{2}+12x+39
- Multiply x(x^{2}+ 2x+ 1) = x^{3}+ 2x^{2}+x
Step 4:: Continue division
- This leaves $$-4x+31
- Multiply 8(x^{2}+ 2x+ 1) = 8x^{2}+ 16x+ 8
Step 5:: Interpret the result
- Divisor: $$x^{2}+2x+1
The division results in:
Step 6:: Verify the form q(x)+\frac{r(x)}{b(x)}
b(x) = x^{2}+2x+1
r(x) = - 4x+ 31
Final Answer
This matches option (A) in the original problem.
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