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:

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


Learn more about this topic:

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Combining Like Terms: Definition, Simplifying & Practice

from High School Algebra I: Help and Review

Chapter 14 / Lesson 8
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