Question

course
ap precalculus
segment
34a - synthetic division and mor
due fri, oct 24, 2025
2
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1. $(x^{4}-5x^{3}+2x^{2}-6x + 3)div(x - 3)$

text math set matrix
answer

Explanation:

Step1: Set up synthetic division

Write the coefficients of the polynomial \(x^{4}-5x^{3}+2x^{2}-6x + 3\) which are \(1,-5,2,-6,3\). The zero of the divisor \(x - 3\) is \(x=3\).

Step2: Bring down the first coefficient

Bring down the first coefficient \(1\).

$$\begin{array}{r|rrrrr} 3&1&-5&2&-6&3\\ & & & & &\\ \hline &1& & & & \end{array}$$

Step3: Multiply and add

Multiply \(3\times1 = 3\), add to \(-5\): \(-5+3=-2\).

$$\begin{array}{r|rrrrr} 3&1&-5&2&-6&3\\ & &3& & &\\ \hline &1&-2& & & \end{array}$$

Multiply \(3\times(-2)=-6\), add to \(2\): \(2+(-6)=-4\).

$$\begin{array}{r|rrrrr} 3&1&-5&2&-6&3\\ & &3&-6& &\\ \hline &1&-2&-4& & \end{array}$$

Multiply \(3\times(-4)=-12\), add to \(-6\): \(-6+(-12)=-18\).

$$\begin{array}{r|rrrrr} 3&1&-5&2&-6&3\\ & &3&-6&-12&\\ \hline &1&-2&-4&-18& \end{array}$$

Multiply \(3\times(-18)=-54\), add to \(3\): \(3+(-54)=-51\).

$$\begin{array}{r|rrrrr} 3&1&-5&2&-6&3\\ & &3&-6&-12&-54\\ \hline &1&-2&-4&-18&-51 \end{array}$$

Step4: Write the quotient and remainder

The quotient polynomial is \(x^{3}-2x^{2}-4x - 18\) and the remainder is \(-51\). So \(\frac{x^{4}-5x^{3}+2x^{2}-6x + 3}{x - 3}=x^{3}-2x^{2}-4x-18-\frac{51}{x - 3}\)

Answer:

\(x^{3}-2x^{2}-4x-18-\frac{51}{x - 3}\)