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Simplify (2x + 5)/(x^2 - 3x) - (3x + 5)/(x^3 - 9x) - (x + 1)/(x^2 - 9)

Sep 3, 2020

#1
+10147
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Simplify (2x + 5)/(x^2 - 3x) - (3x + 5)/(x^3 - 9x) - (x + 1)/(x^2 - 9)

What is the difference?

Hello Gast!

$$\frac{2x+5}{x^2-3x}-\frac{3x+5}{x^3-9x}-\frac{x+1}{x^2-9} \\ =\frac{2x+5}{x(x-3)}-\frac{3x+5}{x(x-3)(x+3) }-\frac{x+1}{(x-3)(x+3)}\\ =\frac{(2x+5)(x+3)-(3x+5)-x(x+1)}{x(x-3)(x+3) }\\ =\frac{2x^2+6x+5x+15-3x-5-x^2-x}{x(x-3)(x+3)}$$

$$=\frac{x^2+7x+10}{x(x-3)(x+3)}\\ =\frac{(x+5)\cdot (x+2)}{x(x-3)(x+3)}\\$$

$$\frac{2x+5}{x^2-3x}-\frac{3x+5}{x^3-9x}-\frac{x+1}{x^2-9}=\color{blue}\frac{(x+5)\cdot (x+2)}{x^3-9x}$$

!

Sep 4, 2020