Questions: Use polynomial long division to rewrite the following fraction in the form q(x)+r(x)/d(x), where d(x) r(x) is the remainder. (x^3+7x^2-18x-6)/(x-2)

$Use polynomial long division to rewrite the following fraction in the form q(x)+r(x)/d(x), where d(x) r(x) is the remainder. (x^3+7x^2-18x-6)/(x-2)$

Transcript text: Use polynomial long division to rewrite the following fraction in the form $q(x)+\frac{r(x)}{d(x)}$, where $d(x)$ $r(x)$ is the remainder. \[ \frac{x^{3}+7 x^{2}-18 x-6}{x-2} \]

Solution

Solution Steps

To solve this problem using polynomial long division, we need to divide the polynomial $x^3 + 7x^2 - 18x - 6$ by $x - 2$. The result will be in the form $q(x) + \frac{r(x)}{d(x)}$, where $q(x)$ is the quotient and $r(x)$ is the remainder.

Step 1: Polynomial Long Division

We start with the polynomial $ P(x) = x^3 + 7x^2 - 18x - 6 $ and divide it by $ D(x) = x - 2 $. Using polynomial long division, we find the quotient and remainder.

Step 2: Finding the Quotient

The quotient obtained from the division is $ Q(x) = x^2 + 9x + 0 $. This simplifies to $ Q(x) = x^2 + 9x $.

Step 3: Finding the Remainder

The remainder from the division is $ R(x) = -6 $.

Final Answer

Thus, we can express the original fraction as: \[ \frac{x^3 + 7x^2 - 18x - 6}{x - 2} = x^2 + 9x + \frac{-6}{x - 2} \] The final result is: \[ \boxed{Q(x) = x^2 + 9x, \quad R(x) = -6} \]

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