Questions: Use synthetic division to find the quotient and the remainder. (x^5+x^3-1) ÷ (x-3) Q(x)= R(x)=

Transcript text: Use synthetic division to find the quotient and the remainder. \[ \begin{array}{l} \left(x^{5}+x^{3}-1\right) \div(x-3) \\ Q(x)=\square \end{array} \] $\square$ \[ R(x)= \] $\square$

Solution

Solution Steps

To solve the problem using synthetic division, we will use the coefficients of the polynomial $x^5 + x^3 - 1$ and divide them by the root of the divisor $x - 3$, which is 3. The process involves bringing down the leading coefficient, multiplying it by the root, adding it to the next coefficient, and repeating this process until all coefficients are processed. The last number obtained is the remainder, and the other numbers form the coefficients of the quotient polynomial.

Step 1: Set Up the Synthetic Division

We are tasked with dividing the polynomial $ P(x) = x^5 + x^3 - 1 $ by $ D(x) = x - 3 $. The coefficients of $ P(x) $ are given as $ [1, 0, 1, 0, 0, -1] $.

Step 2: Perform Synthetic Division

Using synthetic division with the root $ 3 $ from $ D(x) $, we perform the following calculations:

Bring down the leading coefficient $ 1 $.
Multiply $ 1 $ by $ 3 $ and add to the next coefficient $ 0 $ to get $ 3 $.
Multiply $ 3 $ by $ 3 $ and add to the next coefficient $ 1 $ to get $ 10 $.
Multiply $ 10 $ by $ 3 $ and add to the next coefficient $ 0 $ to get $ 30 $.
Multiply $ 30 $ by $ 3 $ and add to the next coefficient $ 0 $ to get $ 90 $.
Finally, multiply $ 90 $ by $ 3 $ and add to the last coefficient $ -1 $ to get the remainder $ 269 $.

The resulting coefficients of the quotient polynomial are $ [1, 3, 10, 30, 90] $.

Step 3: Write the Quotient and Remainder

The quotient polynomial $ Q(x) $ can be expressed as: \[ Q(x) = x^4 + 3x^3 + 10x^2 + 30x + 90 \] The remainder $ R(x) $ is: \[ R(x) = 269 \]