Divide the polynomials. Your answer should be in the form $p(x)+\frac{k}{x-2}$ where $p$ is a polynomial and $k$ is an integer. $\newline$ $\frac{x^{2}-5}{x-2}=$

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Q. Divide the polynomials. Your answer should be in the form

p(x)+\frac{k}{x-2}

where

p

is a polynomial and

k

is an integer.

\newline

\frac{x^{2}-5}{x-2}=

Set up division: Set up the division of the polynomials in long division format. $\newline$ We will divide $(x^2 - 5)$ by $(x - 2)$ using polynomial long division.
Divide first term: Divide the first term of the dividend by the first term of the divisor. $\newline$ Divide $x^2$ by $x$ to get $x$ . This will be the first term of the quotient polynomial $p(x)$ . $\newline$ Calculation: $x^2 / x = x$
Multiply and subtract: Multiply the divisor by the result from Step $2$ and subtract from the dividend. $\newline$ Multiply $(x - 2)$ by $x$ to get $(x^2 - 2x)$ . Subtract this from $(x^2 - 5)$ . $\newline$ Calculation: $(x^2 - 5) - (x^2 - 2x) = 2x - 5$
Bring down next term: Bring down the next term of the dividend, if any. $\newline$ Since there are no more terms to bring down, we proceed to the next step.
Divide result: Divide the result from Step $3$ by the first term of the divisor. $\newline$ Divide $2x$ by $x$ to get $2$ . This will be the next term of the quotient polynomial $p(x)$ . $\newline$ Calculation: $2x / x = 2$
Multiply and subtract: Multiply the divisor by the result from Step $5$ and subtract from the result of Step $3$ . $\newline$ Multiply $(x - 2)$ by $2$ to get $(2x - 4)$ . Subtract this from $(2x - 5)$ . $\newline$ Calculation: $(2x - 5) - (2x - 4) = -1$
Check for completion: Since the degree of the remainder $(-1)$ is less than the degree of the divisor $(x - 2)$ , we cannot continue the division. $\newline$ The remainder is $-1$ , and this will be the value of $k$ in the final answer.
Write final answer: Write the final answer in the form $p(x) + \frac{k}{x - 2}$ . $\newline$ The quotient polynomial $p(x)$ is $x + 2$ , and the remainder is $-1$ . $\newline$ Final Answer: $p(x) = x + 2$ , $k = -1$