Solve (x-1)^(2)/(3x (x -1)^(2)) - (1-2x^(2) 4x)/(x^(3

Question

Simplify the expression

\frac{25 x ^{4} - x ^{3} - 4 x + 1}{3 x ^{5} - 3 x ^{4} - 3 x ^{2} + 3 x}

Evaluate

(x - 1)^{2} \div (3 x (x - 1)^{2}) - (1 - 2 x^{2} \times 4 x) \div ((x^{3} - 1) \times 1) \div (x - 1)

Dividing by a^{n} is the same as multiplying by a^{- n}

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

Multiply

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\frac{( x - 1 ) ^{2} ( x - 1 ) ^{- 2}}{3 x} - (1 - 8 x^{3}) \div ((x^{3} - 1) \times 1) \div (x - 1)

Any expression multiplied by 1 remains the same

\frac{( x - 1 ) ^{2} ( x - 1 ) ^{- 2}}{3 x} - (1 - 8 x^{3}) \div (x^{3} - 1) \div (x - 1)

Multiply the terms

\frac{( x - 1 ) ^{0}}{3 x} - (1 - 8 x^{3}) \div (x^{3} - 1) \div (x - 1)

Rewrite the expression

\frac{( x - 1 ) ^{0}}{3 x} - \frac{1 - 8 x ^{3}}{x ^{3} - 1} \div (x - 1)

Divide the terms

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\frac{( x - 1 ) ^{0}}{3 x} - \frac{1 - 8 x ^{3}}{( x ^{3} - 1 ) ( x - 1 )}

Rewrite the expression

\frac{1}{3 x} - \frac{1 - 8 x ^{3}}{( x ^{3} - 1 ) ( x - 1 )}

Reduce fractions to a common denominator

\frac{( x ^{3} - 1 ) ( x - 1 )}{3 x ( x ^{3} - 1 ) ( x - 1 )} - \frac{( 1 - 8 x ^{3} ) \times 3 x}{( x ^{3} - 1 ) ( x - 1 ) \times 3 x}

Use the commutative property to reorder the terms

\frac{( x ^{3} - 1 ) ( x - 1 )}{3 x ( x ^{3} - 1 ) ( x - 1 )} - \frac{( 1 - 8 x ^{3} ) \times 3 x}{3 ( x ^{3} - 1 ) ( x - 1 ) x}

Rewrite the expression

\frac{( x ^{3} - 1 ) ( x - 1 )}{3 x ( x ^{3} - 1 ) ( x - 1 )} - \frac{( 1 - 8 x ^{3} ) \times 3 x}{3 x ( x ^{3} - 1 ) ( x - 1 )}

Write all numerators above the common denominator

\frac{( x ^{3} - 1 ) ( x - 1 ) - ( 1 - 8 x ^{3} ) \times 3 x}{3 x ( x ^{3} - 1 ) ( x - 1 )}

Multiply the terms

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\frac{x ^{4} - x ^{3} - x + 1 - ( 1 - 8 x ^{3} ) \times 3 x}{3 x ( x ^{3} - 1 ) ( x - 1 )}

Multiply the terms

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\frac{x ^{4} - x ^{3} - x + 1 - ( 3 x - 24 x ^{4} )}{3 x ( x ^{3} - 1 ) ( x - 1 )}

Calculate the sum or difference

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\frac{25 x ^{4} - x ^{3} - 4 x + 1}{3 x ( x ^{3} - 1 ) ( x - 1 )}

Solution

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\frac{25 x ^{4} - x ^{3} - 4 x + 1}{3 x ^{5} - 3 x ^{4} - 3 x ^{2} + 3 x}

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Find the excluded values

x = 0, x = 1

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Find the roots

x_{1} \approx 0.285932, x_{2} \approx 0.411028

Evaluate

(x - 1)^{2} \div (3 x (x - 1)^{2}) - (1 - 2 x^{2} \times 4 x) \div ((x^{3} - 1) \times 1) \div (x - 1)

To find the roots of the expression,set the expression equal to 0

(x - 1)^{2} \div (3 x (x - 1)^{2}) - (1 - 2 x^{2} \times 4 x) \div ((x^{3} - 1) \times 1) \div (x - 1) = 0

Find the domain

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(x - 1)^{2} \div (3 x (x - 1)^{2}) - (1 - 2 x^{2} \times 4 x) \div ((x^{3} - 1) \times 1) \div (x - 1) = 0, x \in (- \infty, 0) \cup (0, 1) \cup (1, + \infty)

Calculate

(x - 1)^{2} \div (3 x (x - 1)^{2}) - (1 - 2 x^{2} \times 4 x) \div ((x^{3} - 1) \times 1) \div (x - 1) = 0

Multiply

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(x - 1)^{2} \div (3 x (x - 1)^{2}) - (1 - 8 x^{3}) \div ((x^{3} - 1) \times 1) \div (x - 1) = 0

Calculate

(x - 1)^{2} \div 3 x (x - 1)^{2} - (1 - 8 x^{3}) \div ((x^{3} - 1) \times 1) \div (x - 1) = 0

Any expression multiplied by 1 remains the same

(x - 1)^{2} \div 3 x (x - 1)^{2} - (1 - 8 x^{3}) \div (x^{3} - 1) \div (x - 1) = 0

Rewrite the expression

(x - 1)^{2} \div 3 x (x - 1)^{2} - \frac{1 - 8 x ^{3}}{x ^{3} - 1} \div (x - 1) = 0

Divide the terms

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\frac{1}{3 x} - \frac{1 - 8 x ^{3}}{x ^{3} - 1} \div (x - 1) = 0

Divide the terms

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\frac{1}{3 x} - \frac{1 - 8 x ^{3}}{( x ^{3} - 1 ) ( x - 1 )} = 0

Subtract the terms

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\frac{25 x ^{4} - x ^{3} - 4 x + 1}{3 x ( x ^{3} - 1 ) ( x - 1 )} = 0

Cross multiply

25 x^{4} - x^{3} - 4 x + 1 = 3 x (x^{3} - 1) (x - 1) \times 0

Simplify the equation

25 x^{4} - x^{3} - 4 x + 1 = 0

Calculate

x \approx 0.411028 x \approx 0.285932

Check if the solution is in the defined range

x \approx 0.411028 x \approx 0.285932, x \in (- \infty, 0) \cup (0, 1) \cup (1, + \infty)

Find the intersection of the solution and the defined range

x \approx 0.411028 x \approx 0.285932

Solution

x_{1} \approx 0.285932, x_{2} \approx 0.411028

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