Questions: (3x^3 + 10x^2 - 12x + 2) ÷ (3x + 1)

Transcript text: $\left.3 x^{3}+10 x^{2}-12 x+2\right) \div(3 x+1)$

Solution

Solution Steps

To solve the polynomial division problem, we will use synthetic division since the divisor is a linear polynomial of the form $3x + 1$. Synthetic division is a simplified form of polynomial division that is particularly useful when dividing by linear factors. We will set up the synthetic division using the coefficients of the dividend polynomial and the root of the divisor.

Step 1: Set Up the Problem

We are tasked with dividing the polynomial $3x^3 + 10x^2 - 12x + 2$ by the linear polynomial $3x + 1$. To perform this division, we will use synthetic division with the root of the divisor.

Step 2: Identify the Root of the Divisor

The divisor $3x + 1$ can be set to zero to find its root: \[ 3x + 1 = 0 \implies x = -\frac{1}{3} \]

Step 3: Perform Synthetic Division

Using synthetic division, we take the coefficients of the dividend polynomial $3, 10, -12, 2$ and perform the division with the root $-\frac{1}{3}$.

Start with the leading coefficient: $3$.
Multiply $3$ by $-\frac{1}{3}$ and add to the next coefficient $10$: \[ 10 + 3 \cdot \left(-\frac{1}{3}\right) = 10 - 1 = 9 \]
Multiply $9$ by $-\frac{1}{3}$ and add to $-12$: \[ -12 + 9 \cdot \left(-\frac{1}{3}\right) = -12 - 3 = -15 \]
Multiply $-15$ by $-\frac{1}{3}$ and add to $2$: \[ 2 + (-15) \cdot \left(-\frac{1}{3}\right) = 2 + 5 = 7 \]

The result of the synthetic division gives us the quotient and the remainder: