Questions: Use synthetic division to rewrite the following fraction in the form q(x)+r(x)/d(x), where d(x) is the denominator of the original fraction, q(x) is the quotient, and r(x) is the remainder. (x^5-5x^4+14x^3-20x^2-12x)/(x-3)

$Use synthetic division to rewrite the following fraction in the form q(x)+r(x)/d(x), where d(x) is the denominator of the original fraction, q(x) is the quotient, and r(x) is the remainder. (x^5-5x^4+14x^3-20x^2-12x)/(x-3)$

Transcript text: Use synthetic division to rewrite the following fraction in the form $q(x)+\frac{r(x)}{d(x)}$, where $d(x)$ is the denominator of the original fraction, $q(x)$ is the quotient, and $r(x)$ is the remainder. \[ \frac{x^{5}-5 x^{4}+14 x^{3}-20 x^{2}-12 x}{x-3} \]

Solution

Solution Steps

To solve the given problem using synthetic division, we will divide the polynomial $x^5 - 5x^4 + 14x^3 - 20x^2 - 12x$ by $x - 3$. Synthetic division is a simplified form of polynomial division, particularly useful when dividing by a linear factor. The process involves using the root of the divisor (in this case, 3) to perform the division, which will yield the quotient polynomial $q(x)$ and the remainder $r(x)$.

Step 1: Perform Synthetic Division

We are given the polynomial $ P(x) = x^5 - 5x^4 + 14x^3 - 20x^2 - 12x $ and we need to divide it by $ d(x) = x - 3 $. Using synthetic division with the root $ 3 $, we find the coefficients of $ P(x) $ as $ [1, -5, 14, -20, -12, 0] $.

Step 2: Calculate the Quotient and Remainder

After performing synthetic division, we obtain the quotient polynomial $ q(x) $ and the remainder $ r $:

Quotient: $ q(x) = x^4 - 2x^3 + 8x^2 + 4 $
Remainder: $ r = 0 $

Step 3: Rewrite the Fraction

The original fraction can now be rewritten in the form: \[ \frac{P(x)}{d(x)} = q(x) + \frac{r}{d(x)} = x^4 - 2x^3 + 8x^2 + 4 + \frac{0}{x - 3} \] Since the remainder is $ 0 $, the fraction simplifies to just the quotient.