 # Best way to simplify a polynomial fraction divided by a polynomial fraction as completely as possible I've been trying for the past few days to complete this question from a review booklet before I start university: Simplify as completely as possible: ( 5x^2 -9x -2 / 30x^3 + 6x^2 ) / ( x^4 -3x^2 -4 / 2x^8 +6x^7 + 4x^6 ) zaviknuogg 2022-09-16 Answered
Best way to simplify a polynomial fraction divided by a polynomial fraction as completely as possible
I've been trying for the past few days to complete this question from a review booklet before I start university:
Simplify as completely as possible:
( 5x^2 -9x -2 / 30x^3 + 6x^2 ) / ( x^4 -3x^2 -4 / 2x^8 +6x^7 + 4x^6 )
However, I've only gotten as far as this answer below:
( (x -1) / 6x^2 ) / ((x^2 +1)(x^2 -4) / (2x^4 +4x^3)(x^4 + x^3))
I can't figure out how to simplify it further. What is the best / a good way to approach such a question that consists of a polynomial fraction divided by a polynomial fraction?
Is it generally a good idea to factor each fraction first then multiply them like I attempted above, or is it better to multiply them without factoring then try to simplify one big fraction?
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$\begin{array}{rl}& \phantom{\rule{thickmathspace}{0ex}}\frac{5{x}^{2}-9x-2}{30{x}^{3}+6{x}^{2}}÷\frac{{x}^{4}-3{x}^{2}-4}{2{x}^{8}+6{x}^{7}+4{x}^{6}}\\ =& \phantom{\rule{thickmathspace}{0ex}}\frac{\left(x-2\right)\left(5x+1\right)}{6{x}^{2}\left(5x+1\right)}×\frac{2{x}^{6}\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)\left({x}^{2}+1\right)}\\ =& \phantom{\rule{thickmathspace}{0ex}}\frac{{x}^{4}\left(x+1\right)}{3\left({x}^{2}+1\right)}\end{array}$
###### Not exactly what you’re looking for? sombereki51
simplifying we obtain
$\frac{\left(5{x}^{2}-9x-2\right)\left(2{x}^{8}+6{x}^{7}+4{x}^{6}\right)}{\left(3{x}^{3}+6{x}^{2}\right)\left({x}^{4}-3{x}^{2}3-4\right)}$
multiplying numerator and denominator out we obtain:
$\frac{10\phantom{\rule{thinmathspace}{0ex}}{x}^{10}+12\phantom{\rule{thinmathspace}{0ex}}{x}^{9}-38\phantom{\rule{thinmathspace}{0ex}}{x}^{8}-48\phantom{\rule{thinmathspace}{0ex}}{x}^{7}-8\phantom{\rule{thinmathspace}{0ex}}{x}^{6}}{3\phantom{\rule{thinmathspace}{0ex}}{x}^{7}+6\phantom{\rule{thinmathspace}{0ex}}{x}^{6}-9\phantom{\rule{thinmathspace}{0ex}}{x}^{5}-18\phantom{\rule{thinmathspace}{0ex}}{x}^{4}-12\phantom{\rule{thinmathspace}{0ex}}{x}^{3}-24\phantom{\rule{thinmathspace}{0ex}}{x}^{2}}$