Question

5.7 divide polynomials: problem 13
1 point
perform the division of polynomials using long division.
\\( \frac { 12 x ^ { 4 } - 17 x ^ { 3 } - 6 x ^ { 2 } + 17 x - 9 } { 4 x - 3 } = \square + \frac { \square } { \square } \\)
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Explanation:

Step1: Divide the leading term of the dividend by the leading term of the divisor

Divide \(12x^{4}\) by \(4x\). \(\frac{12x^{4}}{4x}=3x^{3}\)
Multiply \(3x^{3}\) by \((4x - 3)\): \(3x^{3}(4x - 3)=12x^{4}-9x^{3}\)
Subtract from the dividend: \((12x^{4}-17x^{3}-6x^{2}+17x - 9)-(12x^{4}-9x^{3})=-8x^{3}-6x^{2}+17x - 9\)

Step2: Repeat the process for the new polynomial

Divide \(-8x^{3}\) by \(4x\). \(\frac{-8x^{3}}{4x}=-2x^{2}\)
Multiply \(-2x^{2}\) by \((4x - 3)\): \(-2x^{2}(4x - 3)=-8x^{3}+6x^{2}\)
Subtract: \((-8x^{3}-6x^{2}+17x - 9)-(-8x^{3}+6x^{2})=-12x^{2}+17x - 9\)

Step3: Repeat the process again

Divide \(-12x^{2}\) by \(4x\). \(\frac{-12x^{2}}{4x}=-3x\)
Multiply \(-3x\) by \((4x - 3)\): \(-3x(4x - 3)=-12x^{2}+9x\)
Subtract: \((-12x^{2}+17x - 9)-(-12x^{2}+9x)=8x - 9\)

Step4: One more time

Divide \(8x\) by \(4x\). \(\frac{8x}{4x}=2\)
Multiply \(2\) by \((4x - 3)\): \(2(4x - 3)=8x - 6\)
Subtract: \((8x - 9)-(8x - 6)=-3\)

Answer:

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