Divide the polynomials. $\newline$ The form of your answer should either be $p(x)$ or $p(x)+\frac{k}{x-3}$ where $p(x)$ is a polynomial and $k$ is an integer. $\newline$ $\frac{x^{3}-4 x-15}{x-3}=\square$

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Algebra 1

Powers with decimal and fractional bases

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Q. Divide the polynomials.

\newline

The form of your answer should either be

p(x)

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

where

p(x)

is a polynomial and

k

is an integer.

\newline

\frac{x^{3}-4 x-15}{x-3}=\square

Set up long division: Set up the long division. $\newline$ We will use polynomial long division to divide $(x^3 - 4x - 15)$ by $(x - 3)$ .
Divide first term of dividend by first term of divisor: Divide the first term of the dividend by the first term of the divisor. $\newline$ Divide $x^3$ by $x$ to get $x^2$ . Multiply $(x - 3)$ by $x^2$ and subtract the result from the dividend. $\newline$ $\frac{x^3}{x} = x^2$ $\newline$ $(x - 3) \cdot x^2 = x^3 - 3x^2$
Write down result and bring down next term: Write down the result and bring down the next term. $\newline$ After subtracting, we bring down the next term of the dividend, which is $-4x$ . $\newline$ The new dividend is $-3x^2 - 4x$ .
Divide first term of new dividend by first term of divisor: Divide the first term of the new dividend by the first term of the divisor. $\newline$ Divide $-3x^2$ by $x$ to get $-3x$ . Multiply $(x - 3)$ by $-3x$ and subtract the result from the new dividend. $\newline$ $\frac{-3x^2}{x} = -3x$ $\newline$ $(x - 3) \cdot -3x = -3x^2 + 9x$
Write down result and bring down next term: Write down the result and bring down the next term. $\newline$ After subtracting, we bring down the next term of the dividend, which is $-15$ . $\newline$ The new dividend is $9x - 15$ .
Divide first term of new dividend by first term of divisor: Divide the first term of the new dividend by the first term of the divisor. $\newline$ Divide $9x$ by $x$ to get $9$ . Multiply $(x - 3)$ by $9$ and subtract the result from the new dividend. $\newline$ $\frac{9x}{x} = 9$ $\newline$ $(x - 3) \cdot 9 = 9x - 27$
Write down result and find remainder: Write down the result and find the remainder. $\newline$ After subtracting, we find the remainder. $\newline$ The new dividend is \(-15 - ( $-27$ ) = $12$ ＂). $\newline$ The remainder is \(12＂).
Write final answer: Write the final answer. $\newline$ The quotient is $x^2 - 3x + 9$ with a remainder of $12$ . $\newline$ The final answer is in the form $p(x) + \frac{k}{x - 3}$ , where $p(x)$ is the quotient polynomial and $k$ is the remainder. $\newline$ $p(x) = x^2 - 3x + 9$ $\newline$ $k = 12$