# 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?

• do you mean $$\frac{5x^2-9x-2}{3x^3+6x^2}:\frac{x^4-3x^2-4}{2x^8+6x^7+4x^6}=$$ – Dr. Sonnhard Graubner Jun 10 '17 at 10:14
• @Dr.SonnhardGraubner Yes, sorry, I'm new here and need to learn the math mark up :) – Josh Jun 10 '17 at 10:18

## 2 Answers

\begin{align} &\;\frac{ 5x^2 -9x -2 }{ 30x^3 + 6x^2 } \div \frac{ x^4 -3x^2 -4}{ 2x^8 +6x^7 + 4x^6 }\\ =&\;\frac{(x-2)(5x+1) }{ 6x^2(5x+1) } \times \frac{ 2x^6(x+1)(x+2)}{(x-2)(x+2)(x^2+1)}\\ =&\; \frac{x^4(x+1)}{3(x^2+1)} \end{align}

simplifying we obtain $$\frac{(5x^2-9x-2)(2x^8+6x^7+4x^6)}{(3x^3+6x^2)(x^4-3x^23-4)}$$ multiplying numerator and denominator out we obtain: $$\frac{10\,{x}^{10}+12\,{x}^{9}-38\,{x}^{8}-48\,{x}^{7}-8\,{x}^{6}}{3\,{x}^{7}+6\,{x}^{6}-9\,{x}^{5}-18\,{x}^{4}-12\,{x}^{3}-24\,{x}^{2}}$$