Question

5.7 divide polynomials: problem 7
(1 point)
perform the division of polynomials using long division.
\\( \frac { x ^ { 3 } - 8 x ^ { 2 } + 20 x - 20 } { x - 3 } = \square + \frac { \square } { \square } \\)
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Explanation:

Step1: Divide the leading terms

Divide \(x^{3}\) by \(x\) (the leading term of \(x - 3\)). \(\frac{x^{3}}{x}=x^{2}\).
Multiply \(x - 3\) by \(x^{2}\): \(x^{2}(x - 3)=x^{3}-3x^{2}\).
Subtract \(x^{3}-3x^{2}\) from \(x^{3}-8x^{2}+20x - 20\):
\((x^{3}-8x^{2}+20x - 20)-(x^{3}-3x^{2})=- 5x^{2}+20x - 20\).

Step2: Repeat the process

Divide \(-5x^{2}\) by \(x\): \(\frac{-5x^{2}}{x}=-5x\).
Multiply \(x - 3\) by \(-5x\): \(-5x(x - 3)=-5x^{2}+15x\).
Subtract \(-5x^{2}+15x\) from \(-5x^{2}+20x - 20\):
\((-5x^{2}+20x - 20)-(-5x^{2}+15x)=5x - 20\).

Step3: Final division

Divide \(5x\) by \(x\): \(\frac{5x}{x}=5\).
Multiply \(x - 3\) by \(5\): \(5(x - 3)=5x-15\).
Subtract \(5x - 15\) from \(5x - 20\): \((5x - 20)-(5x - 15)=-5\).

Answer:

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