Questions: Find the quotient using long division. 3x^3 + 2x^2 - 19x + 1 divided by x - 2

Transcript text: Find the quotient using long division. \[ \frac{3 x^{3}+2 x^{2}-19 x+1}{x-2} \]

Solution

Step 1: Divide the Leading Terms

Divide \(3 x^{3}\) by \(x\), resulting in \(3 x^{2}\). The remaining polynomial is \(8 x^{2} - 19 x + 1\).

Step 2: Continue Division

Divide \(8 x^{2}\) by \(x\), resulting in \(8 x\). The remaining polynomial is \(1 - 3 x\).

Step 3: Final Division

Divide \(-3 x\) by \(x\), resulting in \(-3\). The remaining polynomial is \(-5\).

Step 4: Write the Quotient and Remainder

The quotient is \(3 x^{2} + 8 x - 3\) and the remainder is \(-5\).

Step 5: Express the Final Result

The complete division expression is given by: \[ \frac{3 x^{3} + 2 x^{2} - 19 x + 1}{x - 2} = 3 x^{2} + 8 x - 3 - \frac{5}{x - 2} \]

\(\boxed{3x^{2} + 8x - 3 - \frac{5}{x - 2}}\)

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