Questions: Use synthetic division to divide. (12x^3 + 2x^2 - 5x + 1) ÷ (x - 1/3)

Transcript text: Use synthetic division to divide. \[ \left(12 x^{3}+2 x^{2}-5 x+1\right) \div\left(x-\frac{1}{3}\right) \]

Solution

Solution Steps

Step 1: Divide the Leading Terms

Divide \(12 x^{3}\) by \(x\), resulting in \(12 x^{2}\).

Step 2: Multiply and Subtract

Multiply \(12 x^{2}\) by the divisor \(x - \frac{1}{3}\) to get \(12 x^{3} - 4 x^{2}\). Subtract this from the original polynomial: \[ (12 x^{3} + 2 x^{2} - 5 x + 1) - (12 x^{3} - 4 x^{2}) = 6 x^{2} - 5 x + 1 \]

Step 3: Repeat the Process

Now, divide \(6 x^{2}\) by \(x\), resulting in \(6 x\).

Step 4: Multiply and Subtract Again

Multiply \(6 x\) by the divisor \(x - \frac{1}{3}\) to get \(6 x^{2} - 2 x\). Subtract this from the current polynomial: \[ (6 x^{2} - 5 x + 1) - (6 x^{2} - 2 x) = -3 x + 1 \]

Step 5: Final Division

Next, divide \(-3 x\) by \(x\), resulting in \(-3\).

Step 6: Final Multiplication and Subtraction

Multiply \(-3\) by the divisor \(x - \frac{1}{3}\) to get \(-3 x + 1\). Subtract this from the current polynomial: \[ (-3 x + 1) - (-3 x + 1) = 0 \]

Step 7: Write the Quotient and Remainder

The quotient is \(12 x^{2} + 6 x - 3\) and the remainder is \(0\).

Step 8: Final Expression

Thus, the complete division can be expressed as: \[ \frac{12 x^{3} + 2 x^{2} - 5 x + 1}{x - \frac{1}{3}} = 12 x^{2} + 6 x - 3 \]