Question

Answer 1

a: \(\frac{7x-1}{2x^2+6x}=\frac{7x-1}{2x\left(x+3\right)}=\frac{\left(7x-1\right)\left(x-3\right)}{2x\left(x+3\right)\left(x-3\right)}\)

\(=\frac{7x^2-21x-x+3}{2x\left(x-3\right)\left(x+3\right)}=\frac{7x^2-22x+3}{2x\left(x-3\right)\left(x+3\right)}\)

\(\frac{5-3x}{x^2-9}=\frac{5-3x}{\left(x-3\right)\left(x+3\right)}=\frac{2x\left(5-3x\right)}{2x\left(x-3\right)\left(x+3\right)}=\frac{10x-6x^2}{2x\left(x-3\right)\left(x+3\right)}\)

b: \(\frac{x+1}{x-x^2}=\frac{x+1}{-x\left(x-1\right)}=\frac{-\left(x+1\right)}{x\left(x-1\right)}\)

\(=\frac{-\left(x+1\right)\left(x-1\right)}{x\left(x-1\right)^2}=\frac{-x^2+1}{x\left(x-1\right)^2}=\frac{-2x^2+2}{2x\left(x-1\right)^2}\)
\(\frac{x+2}{2-4x+2x^2}=\frac{x+2}{2\left(x^2-2x+1\right)}=\frac{x+2}{2\left(x-1\right)^2}\)

\(=\frac{x\left(x+2\right)}{2x\left(x-1\right)^2}=\frac{x^2+2x}{2x\left(x-1\right)^2}\)

c: \(\frac{4x^2-3x+5}{x^3-1}=\frac{4x^2-3x+5}{\left(x-1\right)\left(x^2+x+1\right)}\)

\(\frac{2x}{x^2+x+1}=\frac{2x\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{2x^2-2x}{\left(x-1\right)\left(x^2+x+1\right)}\)

\(\frac{6}{x-1}=\frac{6\cdot\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}=\frac{6x^2+6x+6}{\left(x-1\right)\left(x^2+x+1\right)}\)

d: \(\frac{7}{5x}=\frac{7\cdot2\cdot\left(x-2y\right)\left(x+2y\right)}{5x\cdot2\cdot\left(x-2y\right)\left(x+2y\right)}=\frac{14x^2-56y^2}{10x\left(x-2y\right)\left(x+2y\right)}\)

\(\frac{4}{x-2y}=\frac{4\cdot5x\cdot\left(x+2y\right)\cdot2}{10x\left(x+2y\right)\left(x-2y\right)}=\frac{40x\left(x+2y\right)}{10x\left(x+2y\right)\left(x-2y\right)}\)

\(\frac{x-y}{8y^2-2x^2}=\frac{-\left(x-y\right)}{2x^2-8y^2}=\frac{-\left(x-y\right)}{2\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{-5x\left(x-y\right)}{2\cdot5x\left(x-2y\right)\left(x+2y\right)}=\frac{-5x^2+5xy}{10x\left(x-2y\right)\left(x+2y\right)}\)

e: \(\frac{5x^2}{x^3+6x^2+12x+8}=\frac{5x^2}{\left(x+2\right)^3}\)

\(\frac{4x}{x^2+4x+4}=\frac{4x}{\left(x+2\right)^2}=\frac{4x\cdot\left(x+2\right)}{\left(x+2\right)^2\cdot\left(x+2\right)}=\frac{4x^2+8x}{\left(x+2\right)^3}\)