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Question: Use polynomial long division to rewrite the following fraction in the form \( q(x) +…

Use polynomial long division to rewrite the following fraction in the form q(x)+r(x)d(x) q(x) + \frac{r(x)}{d(x)} where d(x) d(x) is the denominator of the original fraction, the quotient, and r(x) r(x) is the remainder.

4x34x2+9x+62x+1 \frac{4x^3 - 4x^2 + 9x + 6}{2x + 1}

Answer: 6 Points

2x23x+6 2x^2 - 3x + 6

Solution

Step 1: 4x34x2+9x+62x+1 \frac{4x^3 - 4x^2 + 9x + 6}{2x + 1} Step 2: 4x3÷2x=2x2 4x^3 \div 2x = 2x^2 Step 3: 2x2×(2x+1)=4x3+2x2 2x^2 \times (2x + 1) = 4x^3 + 2x^2 Step 4: (4x34x2+9x+6)(4x3+2x2)=6x2+9x+6 (4x^3 - 4x^2 + 9x + 6) - (4x^3 + 2x^2) = -6x^2 + 9x + 6 Step 5: 6x2÷2x=3x -6x^2 \div 2x = -3x Step 6: 3x×(2x+1)=6x23x -3x \times (2x + 1) = -6x^2 - 3x Step 7: (6x2+9x+6)(6x23x)=12x+6 (-6x^2 + 9x + 6) - (-6x^2 - 3x) = 12x + 6 Step 8: 12x÷2x=6 12x \div 2x = 6 Step 9: 6×(2x+1)=12x+6 6 \times (2x + 1) = 12x + 6 Step 10: (12x+6)(12x+6)=0 (12x + 6) - (12x + 6) = 0 Final Answer: 2x23x+6 2x^2 - 3x + 6

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