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Chapter 7: Problem 58

Divide as indicated. $$\frac{3 y+12}{y^{2}+3 y} \div \frac{y^{2}+y-12}{9 y-y^{3}}$$

Short Answer

Expert verified
\[\frac{-3 y^{3}-12 y^{2}+27 y+108}{y^{3}+4y^{2}-21y-36}\]

Step by step solution

01

Rewrite the Division as Multiplication

By the 'keep, change, flip' rule, \[\frac{3 y+12}{y^{2}+3 y} \div \frac{y^{2}+y-12}{9 y-y^{3}} \]can be rewritten as\[\frac{3 y+12}{y^{2}+3 y} \cdot \frac{9 y-y^{3}}{y^{2}+y-12}\]
02

Multiply the Fractions

We then multiply the numerators together and the denominators together to form the new fraction:\[\frac{(3 y+12)(9 y-y^{3})}{(y^{2}+3 y)(y^{2}+y-12)}\]
03

Simplify the Fractions

Now, simplify the expression and factor where possible. The numerator becomes: \[27 y^{2}-3 y^{4}+108 y-12 y^{3}\]Rearrange them to match the standard polynomial format \(a x^{n}+b x^{n-1}+\ldots+k\):\[ -3 y^{4}-12 y^{3}+27 y^{2}+108 y\]The denominator becomes:\[y^{4}+4y^{3}+3y^{2}-12y^{2}-36y-12y^{2}\]Combine the like terms\[y^{4}+4y^{3}-21y^{2}-36y\]
04

Find the Greatest commom factor (GCF)

From the numerator, we can factor out y to get\[ y(-3 y^{3}-12 y^{2}+27 y+108)\]From the denominator, we can factor out y too to get\[ y(y^{3}+4y^{2}-21y-36)\]Here the greatest common factor in both numerator and denominator is y. After cancelling it out, the expression becomes\[\frac{-3 y^{3}-12 y^{2}+27 y+108}{y^{3}+4y^{2}-21y-36}\]
05

Final Simplification

The above expression is the final simplified result. It cannot be simplified any further.

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Most popular questions from this chapter

add or subtract as indicated. Simplify the result, if possible. $$\begin{aligned} &\frac{x^{2}+9 x}{4 x^{2}-11 x-3}+\frac{3 x-5 x^{2}}{4 x^{2}-11 x-3}\\\ &x^{2}-4 x-4 x-4 \end{aligned}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{6 x+5}{x-2}+\frac{4 x}{2-x}$$

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