Questions: #16. (3x^3 - 4x^2 - 23x) = (x - 2)

Transcript text: #16. (3x^3 - 4x^2 - 23x) = (x - 2)

Solution

Solution Steps

Step 1: Set Up the Polynomial Division

We need to divide the polynomial \(3x^3 - 4x^2 - 23x\) by \(x - 2\). This can be done using polynomial long division.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(3x^3\) by the leading term of the divisor \(x\): \[ \frac{3x^3}{x} = 3x^2 \]

Step 3: Multiply and Subtract

Multiply \(3x^2\) by \(x - 2\) and subtract the result from the original polynomial: \[ (3x^3 - 4x^2 - 23x) - (3x^3 - 6x^2) = (3x^3 - 4x^2 - 23x) - 3x^3 + 6x^2 = 2x^2 - 23x \]

Step 4: Repeat the Process

Divide the new leading term \(2x^2\) by the leading term of the divisor \(x\): \[ \frac{2x^2}{x} = 2x \]

Step 5: Multiply and Subtract Again

Multiply \(2x\) by \(x - 2\) and subtract the result from the new polynomial: \[ (2x^2 - 23x) - (2x^2 - 4x) = (2x^2 - 23x) - 2x^2 + 4x = -19x \]

Step 6: Final Division

Divide the new leading term \(-19x\) by the leading term of the divisor \(x\): \[ \frac{-19x}{x} = -19 \]

Step 7: Multiply and Subtract One Last Time

Multiply \(-19\) by \(x - 2\) and subtract the result from the new polynomial: \[ (-19x) - (-19x + 38) = -19x + 19x - 38 = -38 \]

Final Answer

The quotient is \(3x^2 + 2x - 19\) and the remainder is \(-38\). Therefore, the result of the division is: \[ 3x^2 + 2x - 19 + \frac{-38}{x-2} \]

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