What is expand 3/(x+2)-2/(3x-1)-4/((3x-1)^2) ?

The solution to expand 3/(x+2)-2/(3x-1)-4/((3x-1)^2) is (21x^2-32x-1)/(9x^3+12x^2-11x+2)

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expand 3/(x+2)-2/(3x-1)-4/((3x-1)^2)

expand

\frac{3}{x + 2} - \frac{2}{3 x - 1} - \frac{4}{( 3 x - 1 ) ^{2}}

\frac{21 x ^{2} - 32 x - 1}{9 x ^{3} + 12 x ^{2} - 11 x + 2}

Solution steps

\frac{3}{x + 2} - \frac{2}{3 x - 1} - \frac{4}{( 3 x - 1 ) ^{2}}

(3 x - 1)^{2} = 9 x^{2} - 6 x + 1

= \frac{3}{x + 2} - \frac{2}{3 x - 1} - \frac{4}{9 x ^{2} - 6 x + 1}

Factor

9 x^{2} - 6 x + 1 : (3 x - 1)^{2}

= \frac{3}{x + 2} - \frac{2}{3 x - 1} - \frac{4}{( 3 x - 1 ) ^{2}}

Least Common Multiplier of

x + 2, 3 x - 1, (3 x - 1)^{2} : (x + 2) (3 x - 1)^{2}

Adjust Fractions based on the LCM

= \frac{3 ( 3 x - 1 ) ^{2}}{( x + 2 ) ( 3 x - 1 ) ^{2}} - \frac{2 ( x + 2 ) ( 3 x - 1 )}{( x + 2 ) ( 3 x - 1 ) ^{2}} - \frac{4 ( x + 2 )}{( 3 x - 1 ) ^{2} ( x + 2 )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 ( 3 x - 1 ) ^{2} - 2 ( x + 2 ) ( 3 x - 1 ) - 4 ( x + 2 )}{( x + 2 ) ( 3 x - 1 ) ^{2}}

(3 x - 1)^{2} = 9 x^{2} - 6 x + 1

= \frac{3 ( 3 x - 1 ) ^{2} - 2 ( x + 2 ) ( 3 x - 1 ) - 4 ( x + 2 )}{( x + 2 ) ( 9 x ^{2} - 6 x + 1 )}

(3 x - 1)^{2} = 9 x^{2} - 6 x + 1

= \frac{3 ( 9 x ^{2} - 6 x + 1 ) - 2 ( x + 2 ) ( 3 x - 1 ) - 4 ( x + 2 )}{( x + 2 ) ( 9 x ^{2} - 6 x + 1 )}

Expand

(x + 2) (9 x^{2} - 6 x + 1) : 9 x^{3} + 12 x^{2} - 11 x + 2

= \frac{3 ( 9 x ^{2} - 6 x + 1 ) - 2 ( x + 2 ) ( 3 x - 1 ) - 4 ( x + 2 )}{9 x ^{3} + 12 x ^{2} - 11 x + 2}

Expand

3 (9 x^{2} - 6 x + 1) - 2 (x + 2) (3 x - 1) - 4 (x + 2) : 21 x^{2} - 32 x - 1

= \frac{21 x ^{2} - 32 x - 1}{9 x ^{3} + 12 x ^{2} - 11 x + 2}

e^{x} (e^{x} + x \cdot e^{x})

\frac{( a ( x + h ) + b ) - ( a x + b )}{h}

50 (\frac{12 sin ( \frac{π \cdot x}{12} )}{π} + x)

\frac{d}{d x} ((x - y^{3})^{2} - x^{3})

(y = x^{2} + 2 x) (x + 2)

What is expand 3/(x+2)-2/(3x-1)-4/((3x-1)^2) ?
The solution to expand 3/(x+2)-2/(3x-1)-4/((3x-1)^2) is (21x^2-32x-1)/(9x^3+12x^2-11x+2)