zaviknuogg

2022-09-16

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?

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?

Rayna Aguilar

Beginner2022-09-17Added 14 answers

$\begin{array}{rl}& \phantom{\rule{thickmathspace}{0ex}}\frac{5{x}^{2}-9x-2}{30{x}^{3}+6{x}^{2}}\xf7\frac{{x}^{4}-3{x}^{2}-4}{2{x}^{8}+6{x}^{7}+4{x}^{6}}\\ =& \phantom{\rule{thickmathspace}{0ex}}\frac{(x-2)(5x+1)}{6{x}^{2}(5x+1)}\times \frac{2{x}^{6}(x+1)(x+2)}{(x-2)(x+2)({x}^{2}+1)}\\ =& \phantom{\rule{thickmathspace}{0ex}}\frac{{x}^{4}(x+1)}{3({x}^{2}+1)}\end{array}$

sombereki51

Beginner2022-09-18Added 3 answers

simplifying we obtain

$\frac{(5{x}^{2}-9x-2)(2{x}^{8}+6{x}^{7}+4{x}^{6})}{(3{x}^{3}+6{x}^{2})({x}^{4}-3{x}^{2}3-4)}$

multiplying numerator and denominator out we obtain:

$\frac{10\phantom{\rule{thinmathspace}{0ex}}{x}^{10}+12\phantom{\rule{thinmathspace}{0ex}}{x}^{9}-38\phantom{\rule{thinmathspace}{0ex}}{x}^{8}-48\phantom{\rule{thinmathspace}{0ex}}{x}^{7}-8\phantom{\rule{thinmathspace}{0ex}}{x}^{6}}{3\phantom{\rule{thinmathspace}{0ex}}{x}^{7}+6\phantom{\rule{thinmathspace}{0ex}}{x}^{6}-9\phantom{\rule{thinmathspace}{0ex}}{x}^{5}-18\phantom{\rule{thinmathspace}{0ex}}{x}^{4}-12\phantom{\rule{thinmathspace}{0ex}}{x}^{3}-24\phantom{\rule{thinmathspace}{0ex}}{x}^{2}}$

$\frac{(5{x}^{2}-9x-2)(2{x}^{8}+6{x}^{7}+4{x}^{6})}{(3{x}^{3}+6{x}^{2})({x}^{4}-3{x}^{2}3-4)}$

multiplying numerator and denominator out we obtain:

$\frac{10\phantom{\rule{thinmathspace}{0ex}}{x}^{10}+12\phantom{\rule{thinmathspace}{0ex}}{x}^{9}-38\phantom{\rule{thinmathspace}{0ex}}{x}^{8}-48\phantom{\rule{thinmathspace}{0ex}}{x}^{7}-8\phantom{\rule{thinmathspace}{0ex}}{x}^{6}}{3\phantom{\rule{thinmathspace}{0ex}}{x}^{7}+6\phantom{\rule{thinmathspace}{0ex}}{x}^{6}-9\phantom{\rule{thinmathspace}{0ex}}{x}^{5}-18\phantom{\rule{thinmathspace}{0ex}}{x}^{4}-12\phantom{\rule{thinmathspace}{0ex}}{x}^{3}-24\phantom{\rule{thinmathspace}{0ex}}{x}^{2}}$

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