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Math  /  Algebra

QuestionQuestion Watch Video Show Examples Use synthetic division to find the result when 3x3+16x2+14x−83 x^{3}+16 x^{2}+14 x-8 is divided by x+4x+4.

Studdy Solution

STEP 1

1. We are given the polynomial 3x3+16x2+14x−8 3x^3 + 16x^2 + 14x - 8 .
2. We need to divide this polynomial by x+4 x + 4 using synthetic division.

STEP 2

1. Set up the synthetic division.
2. Perform the synthetic division process.
3. Interpret the result of the synthetic division.

STEP 3

To set up synthetic division, we need the coefficients of the polynomial and the root of the divisor. The coefficients of the polynomial 3x3+16x2+14x−8 3x^3 + 16x^2 + 14x - 8 are 3,16,14,−8 3, 16, 14, -8 .
The divisor is x+4 x + 4 , so the root is −4 -4 .

STEP 4

Write the root −4 -4 on the left, and the coefficients 3,16,14,−8 3, 16, 14, -8 on the right:
−431614−8\begin{array}{c|cccc} -4 & 3 & 16 & 14 & -8 \\ & & & & \\ \hline & & & & \\ \end{array}
Bring down the leading coefficient 3 3 to the bottom row:
−431614−83\begin{array}{c|cccc} -4 & 3 & 16 & 14 & -8 \\ & & & & \\ \hline & 3 & & & \\ \end{array}

STEP 5

Multiply the root −4 -4 by the number just written on the bottom row 3 3 , and write the result under the next coefficient:
−431614−8−123\begin{array}{c|cccc} -4 & 3 & 16 & 14 & -8 \\ & & -12 & & \\ \hline & 3 & & & \\ \end{array}
Add the column:
−431614−8−1234\begin{array}{c|cccc} -4 & 3 & 16 & 14 & -8 \\ & & -12 & & \\ \hline & 3 & 4 & & \\ \end{array}

STEP 6

Repeat the process: Multiply the root −4 -4 by the last number written on the bottom row 4 4 , and write the result under the next coefficient:
−431614−8−12−1634\begin{array}{c|cccc} -4 & 3 & 16 & 14 & -8 \\ & & -12 & -16 & \\ \hline & 3 & 4 & & \\ \end{array}
Add the column:
−431614−8−12−1634−2\begin{array}{c|cccc} -4 & 3 & 16 & 14 & -8 \\ & & -12 & -16 & \\ \hline & 3 & 4 & -2 & \\ \end{array}

STEP 7

Repeat the process one more time: Multiply the root −4 -4 by the last number written on the bottom row −2 -2 , and write the result under the next coefficient:
−431614−8−12−16834−2\begin{array}{c|cccc} -4 & 3 & 16 & 14 & -8 \\ & & -12 & -16 & 8 \\ \hline & 3 & 4 & -2 & \\ \end{array}
Add the column:
−431614−8−12−16834−20\begin{array}{c|cccc} -4 & 3 & 16 & 14 & -8 \\ & & -12 & -16 & 8 \\ \hline & 3 & 4 & -2 & 0 \\ \end{array}

STEP 8

The numbers on the bottom row represent the coefficients of the quotient polynomial. Since the remainder is 0 0 , the division is exact.
The quotient is 3x2+4x−2 3x^2 + 4x - 2 .
The result of the division is:
3x2+4x−2 3x^2 + 4x - 2

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