Divide the polynomials. $\newline$ Your answer should be in the form $p(x)+\frac{k}{x-3}$ where $p$ is a polynomial and $k$ is an integer. $\newline$ $\frac{x^{2}+4}{x-3}=$

Full solution

Q. Divide the polynomials.

\newline

Your answer should be in the form

p(x)+\frac{k}{x-3}

where

p

is a polynomial and

k

is an integer.

\newline

\frac{x^{2}+4}{x-3}=

Set up division: Set up the division of the polynomials using long division. $\newline$ We are dividing $x^2 + 4$ by $x - 3$ . We will use polynomial long division to find the quotient and the remainder.
Divide first terms: Divide the first term of the dividend by the first term of the divisor. Divide $x^2$ by $x$ to get $x$ . This will be the first term of the quotient polynomial $p(x)$ .
Subtract and find new dividend: Multiply the divisor by the term found in Step $2$ and subtract from the dividend. $\newline$ Multiply $(x - 3)$ by $x$ to get $x^2 - 3x$ . Subtract this from $x^2 + 4$ to find the new dividend. $\newline$ $(x^2 + 4) - (x^2 - 3x) = 3x + 4$ .
Divide new dividend: Divide the new dividend by the first term of the divisor. $\newline$ Divide $3x$ by $x$ to get $3$ . This will be the next term of the quotient polynomial $p(x)$ .
Subtract and find remainder: Multiply the divisor by the term found in Step $4$ and subtract from the new dividend. $\newline$ Multiply $(x - 3)$ by $3$ to get $3x - 9$ . Subtract this from $3x + 4$ to find the remainder. $\newline$ $(3x + 4) - (3x - 9) = 13$ .
Write final answer: Write the final answer. $\newline$ The quotient polynomial $p(x)$ is $x + 3$ and the remainder is $13$ . Therefore, the final answer in the form $p(x) + \frac{k}{x - 3}$ is: $\newline$ $x + 3 + \frac{13}{x - 3}$ .