QuestionAugust 24, 2025

Divide the polynomials. The form of your answer should either be p(x)orp(x)+(k)/(x-2) where p(x) is a polynomial and kis an integer. (x^3+6x^2-5x)/(x-2)=square

Divide the polynomials. The form of your answer should either be p(x)orp(x)+(k)/(x-2) where p(x) is a polynomial and kis an integer. (x^3+6x^2-5x)/(x-2)=square
Divide the polynomials. The form of your answer should either be p(x)orp(x)+(k)/(x-2) where p(x) is a polynomial and kis an integer. (x^3+6x^2-5x)/(x-2)=square

Solution
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Answer

x^2 + 8x + 11 + \frac{22}{x-2} Explanation 1. Perform Polynomial Long Division Divide x^3 + 6x^2 - 5x by x - 2. Start with the leading term: \frac{x^3}{x} = x^2. Multiply x^2 by x - 2: x^3 - 2x^2. Subtract from the original polynomial: (x^3 + 6x^2 - 5x) - (x^3 - 2x^2) = 8x^2 - 5x. 2. Continue Division Process Divide 8x^2 by x: 8x. Multiply 8x by x - 2: 8x^2 - 16x. Subtract: (8x^2 - 5x) - (8x^2 - 16x) = 11x. 3. Finalize Division Divide 11x by x: 11. Multiply 11 by x - 2: 11x - 22. Subtract: (11x) - (11x - 22) = 22. 4. Write the Result The quotient is x^2 + 8x + 11 and the remainder is 22. Express the division as x^2 + 8x + 11 + \frac{22}{x-2}.

Explanation

1. Perform Polynomial Long Division<br /> Divide $x^3 + 6x^2 - 5x$ by $x - 2$. Start with the leading term: $\frac{x^3}{x} = x^2$. Multiply $x^2$ by $x - 2$: $x^3 - 2x^2$. Subtract from the original polynomial: $(x^3 + 6x^2 - 5x) - (x^3 - 2x^2) = 8x^2 - 5x$.<br />2. Continue Division Process<br /> Divide $8x^2$ by $x$: $8x$. Multiply $8x$ by $x - 2$: $8x^2 - 16x$. Subtract: $(8x^2 - 5x) - (8x^2 - 16x) = 11x$.<br />3. Finalize Division<br /> Divide $11x$ by $x$: $11$. Multiply $11$ by $x - 2$: $11x - 22$. Subtract: $(11x) - (11x - 22) = 22$.<br />4. Write the Result<br /> The quotient is $x^2 + 8x + 11$ and the remainder is $22$. Express the division as $x^2 + 8x + 11 + \frac{22}{x-2}$.
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