Question

use synthetic division to find the quotient and remainder when $-x^4 + 6x^3 + 3x - 9$ is divided by $x - 6$ by completing the parts below. (a) complete this synthetic division table. \\(6\overline{)\\ -1\\ \\ 6\\ \\ 0\\ \\ 3\\ -9}\\) (b) write your answer in the following form: quotient + \\(\frac{\text{remainder}}{x - 6}\\). \\(\frac{-x^4 + 6x^3 + 3x - 9}{x - 6} = \square + \frac{\square}{x - 6}\\)

Explanation:

Step1: Recall Synthetic Division Rules

Synthetic division is used for dividing a polynomial by a linear factor \(x - c\). Here, \(c = 6\), and the polynomial is \(-x^{4}+6x^{3}+0x^{2}+3x - 9\) (we include the \(0x^{2}\) term for the missing degree). The coefficients are \(-1, 6, 0, 3, -9\).

Step2: Perform Synthetic Division

• Bring down the first coefficient: \(-1\).
• Multiply by \(c = 6\): \(-1\times6=-6\). Add to the next coefficient: \(6+(-6) = 0\).
• Multiply \(0\) by \(6\): \(0\times6 = 0\). Add to the next coefficient: \(0 + 0=0\).
• Multiply \(0\) by \(6\): \(0\times6 = 0\). Add to the next coefficient: \(3+0 = 3\).
• Multiply \(3\) by \(6\): \(3\times6 = 18\). Add to the last coefficient: \(-9 + 18=9\).

So the synthetic division table (the top row of boxes is the products, the bottom row is the sums) is:
Top row (products): \(-6, 0, 0, 18\)
Bottom row (sums): \(-1, 0, 0, 3, 9\)

Step3: Determine Quotient and Remainder

The remainder is the last number in the bottom row, \(9\). The quotient is a polynomial of degree one less than the original, with coefficients from the bottom row (excluding the remainder). So the quotient is \(-x^{3}+0x^{2}+0x + 3=-x^{3}+3\).

Step4: Write in the Required Form

The form is \(\text{Quotient}+\frac{\text{Remainder}}{x - 6}\), so \(\frac{-x^{4}+6x^{3}+3x - 9}{x - 6}=-x^{3}+3+\frac{9}{x - 6}\).

Answer:

(a) The synthetic division table (top row of products: \(-6, 0, 0, 18\); bottom row of sums: \(-1, 0, 0, 3, 9\))
(b) \(-x^{3}+3+\frac{9}{x - 6}\) (so the quotient box is \(-x^{3}+3\) and the remainder box is \(9\))