# Simplify. \frac{x^{6}-13x^{3}+42}{x^{3}-6}

## Question:

Simplify.

{eq}\displaystyle \frac{x^{6} - 13x^{3} + 42}{x^{3} - 6} {/eq}

## Simplification:

Simplification is a method of getting the most basic form of a complex expression by applying different mathematical techniques. To simplify the given problem, we have here used common division and multiplications rules to extract the given problem to an irreducible form.

## Answer and Explanation:

Given: $$\displaystyle \dfrac{x^{6} - 13x^{3} + 42}{x^{3} - 6} $$

Let's divide the leading coefficients of the numerator : {eq}x^{6} - 13x^{3} + 42 {/eq} and the divisor {eq}\displaystyle x^{3} - 6 {/eq}:

$$\begin{align} &=\dfrac {x^{6}}{x^{3}} \\[0.3cm] &= x^{3} \end{align} $$

Therefore, the quotient is {eq}=x^{3} {/eq}. Now, let's multiply {eq}x^{3} - 6 {/eq} by {eq}x^{3} {/eq}:

$$=x^{6} - 6x^{3} $$

Let's subtract {eq}x^{6} - 6x^{3} {/eq} from {eq}x^{6} - 13x^{3} + 42 {/eq} to get the remainder.

$$\begin{align} &= (x^{6} - 13x^{3} + 42) - (x^{6} - 6x^{3}) \\[0.3cm] &= x^{6} - 13x^{3} + 42 - x^{6} + 6x^{3} \\[0.3cm] &= -7x^{3} + 42 \end{align} $$

Combining the above, the given expression is simplified to:

$$\displaystyle \dfrac{x^{6} - 13x^{3} + 42}{x^{3} - 6} = x^{3} + \dfrac{ -7x^{3} + 42}{x^{3} - 6} $$

Factoring out {eq}x^3 - 6 {/eq} in the second term:

$$\begin{align} \dfrac{ -7x^{3} + 42}{x^{3} - 6} &=\dfrac{-7(x^{3} - 6)}{(x^{3} - 6)} \\[0.3cm] &= -7 \end{align} $$

Therefore, the simplified form is:

$$= \boxed{\displaystyle x^{3} - 7} $$

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from High School Algebra I: Help and Review

Chapter 14 / Lesson 8