Questions: Use synthetic division to find the result when x^4-10x^2-13x-50 is divided by x+5. Write your answer in the form q(x)+r(x)/d(x), where q(x) is the quotient, r(x) is the remainder, and d(x) is the divisor.

Use synthetic division to find the result when x^4-10x^2-13x-50 is divided by x+5. Write your answer in the form q(x)+r(x)/d(x), where q(x) is the quotient, r(x) is the remainder, and d(x) is the divisor.

Transcript text: Question Use synthetic division to find the result when $x^{4}-10 x^{2}-13 x-50$ is divided by $x+5$. Write your answer in the form $q(x)+\frac{r(x)}{d(x)}$, where $q(x)$ is the quotient, $r(x)$ is the remainder, and $d(x)$ is the divisor. Provide your answer below:

Solution

Solution Steps

Hint

To solve the problem using synthetic division, first identify the root of the divisor and then apply the synthetic division process by iteratively multiplying the root by the current result and adding it to the next coefficient to find the coefficients of the quotient polynomial and the remainder.

Step 1: Identify the Root of the Divisor

The divisor is $ x + 5 $. The root of the divisor is $ -5 $.

Step 2: Set Up Synthetic Division

The coefficients of the polynomial $ x^4 - 10x^2 - 13x - 50 $ are: \[ [1, 0, -10, -13, -50] \]

Step 3: Perform Synthetic Division

Using the root $ -5 $, we perform synthetic division:

Bring down the leading coefficient: \[ 1 \]
Multiply the root by the current result and add to the next coefficient: \[ 1 \cdot (-5) + 0 = -5 \] \[ -5 \cdot (-5) + (-10) = 25 - 10 = 15 \] \[ 15 \cdot (-5) + (-13) = -75 - 13 = -88 \] \[ -88 \cdot (-5) + (-50) = 440 - 50 = 390 \]

The new coefficients are: \[ [1, -5, 15, -88, 390] \]

Step 4: Identify Quotient and Remainder

The quotient polynomial $ q(x) $ is formed by the first four coefficients: \[ q(x) = 1x^3 - 5x^2 + 15x - 88 \]

The remainder $ r(x) $ is the last coefficient: \[ r(x) = 390 \]

The divisor $ d(x) $ remains: \[ d(x) = x + 5 \]