Question

Decompose the following rational fractions into simple rational fractionsA) X²-x-2/x⁴-5x²+4B)8³−7−1/⁴−4² + 3

Answer 1

A)

`\begin{gathered} (x^2-x-2)/(x^4-5x^2+4) \\ factor\text{ }x^2-x-2 \\ (\left(x+1\right)\left(x-2\right))/(x^4-5x^2+4) \\ factor\text{ }x^4-5x^2+4 \\ (\left(x+1\right)\left(x-2\right))/(\left(x+1\right)\left(x-1\right)\left(x+2\right)\left(x-2\right)) \\ We\text{ remove common terms }x+1 \\ (x-2)/(\left(x-1\right)\left(x+2\right)\left(x-2\right)) \\ We\text{ remove common terms x-2} \\ (1)/((x-1)(x+2)) \end{gathered}`

B)

`\begin{gathered} (8x^3-7x-1)/(x^4-4x^2+3) \\ \text{factor }8x^3-7x-1 \\ (\left(x-1\right)\left(8x^2+8x+1\right))/(x^4-4x^2+3) \\ \text{factor }x^4-4x^2+3 \\ (\left(x-1\right)\left(8x^2+8x+1\right))/(\left(x+1\right)\left(x-1\right)\left(x^2-3\right)) \\ We\text{ remove common terms x-1} \\ (8x^2+8x+1)/(\left(x+1\right)\left(x^2-3\right)) \end{gathered}`