Solve (2x^5 2/x 1)(x^2 -1/ x^2)

Question

Simplify the expression

4 x^{6} - 4 x^{2}

Evaluate

(2 x^{5} \times \frac{2}{x} \times 1) (x^{2} - \frac{1}{x ^{2}})

Remove the parentheses

2 x^{5} \times \frac{2}{x} \times 1 \times (x^{2} - \frac{1}{x ^{2}})

Subtract the terms

More Steps

Simplify

x^{2} - \frac{1}{x ^{2}}

Reduce fractions to a common denominator

\frac{x ^{2} \times x ^{2}}{x ^{2}} - \frac{1}{x ^{2}}

Write all numerators above the common denominator

\frac{x ^{2} \times x ^{2} - 1}{x ^{2}}

Multiply the terms

More Steps

Evaluate

x^{2} \times x^{2}

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

x^{2 + 2}

Add the numbers

x^{4}

\frac{x ^{4} - 1}{x ^{2}}

2 x^{5} \times \frac{2}{x} \times 1 \times \frac{x ^{4} - 1}{x ^{2}}

Rewrite the expression

2 x^{5} \times \frac{2}{x} \times \frac{x ^{4} - 1}{x ^{2}}

Multiply the terms

More Steps

Multiply the terms

2 x^{5} \times \frac{2}{x}

Cancel out the common factor x

2 x^{4} \times 2

Multiply the terms

4 x^{4}

4 x^{4} \times \frac{x ^{4} - 1}{x ^{2}}

Cancel out the common factor x^{2}

4 x^{2} (x^{4} - 1)

Apply the distributive property

4 x^{2} \times x^{4} - 4 x^{2} \times 1

Multiply the terms

More Steps

Evaluate

x^{2} \times x^{4}

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

x^{2 + 4}

Add the numbers

x^{6}

4 x^{6} - 4 x^{2} \times 1

Solution

4 x^{6} - 4 x^{2}

Show Solution

Find the excluded values

x = 0

Evaluate

(2 x^{5} \times \frac{2}{x} \times 1) (x^{2} - \frac{1}{x ^{2}})

To find the excluded values,set the denominators equal to 0

x = 0 x^{2} = 0

The only way a power can be 0 is when the base equals 0

x = 0 x = 0

Solution

x = 0

Show Solution

Factor the expression

4 x^{2} (x - 1) (x + 1) (x^{2} + 1)

Evaluate

(2 x^{5} \times \frac{2}{x} \times 1) (x^{2} - \frac{1}{x ^{2}})

Remove the parentheses

2 x^{5} \times \frac{2}{x} \times 1 \times (x^{2} - \frac{1}{x ^{2}})

Subtract the terms

More Steps

Simplify

x^{2} - \frac{1}{x ^{2}}

Reduce fractions to a common denominator

\frac{x ^{2} \times x ^{2}}{x ^{2}} - \frac{1}{x ^{2}}

Write all numerators above the common denominator

\frac{x ^{2} \times x ^{2} - 1}{x ^{2}}

Multiply the terms

More Steps

Evaluate

x^{2} \times x^{2}

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

x^{2 + 2}

Add the numbers

x^{4}

\frac{x ^{4} - 1}{x ^{2}}

2 x^{5} \times \frac{2}{x} \times 1 \times \frac{x ^{4} - 1}{x ^{2}}

Multiply the terms

More Steps

Multiply the terms

2 x^{5} \times \frac{2}{x} \times 1

Rewrite the expression

2 x^{5} \times \frac{2}{x}

Cancel out the common factor x

2 x^{4} \times 2

Multiply the terms

4 x^{4}

4 x^{4} \times \frac{x ^{4} - 1}{x ^{2}}

Cancel out the common factor x^{2}

4 x^{2} (x^{4} - 1)

Solution

More Steps

Evaluate

x^{4} - 1

Use a^{2} - b^{2} = (a - b) (a + b) to factor the expression

(x^{2} - 1) (x^{2} + 1)

Use a^{2} - b^{2} = (a - b) (a + b) to factor the expression

(x - 1) (x + 1) (x^{2} + 1)

4 x^{2} (x - 1) (x + 1) (x^{2} + 1)

Show Solution

Find the roots

x_{1} = - 1, x_{2} = 1

Evaluate

(2 x^{5} \times \frac{2}{x} \times 1) (x^{2} - \frac{1}{x ^{2}})

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

(2 x^{5} \times \frac{2}{x} \times 1) (x^{2} - \frac{1}{x ^{2}}) = 0

Find the domain

More Steps

Evaluate

{x \neq = 0 x^{2} \neq = 0

The only way a power can not be 0 is when the base not equals 0

{x \neq = 0 x \neq = 0

Find the intersection

x \neq = 0

(2 x^{5} \times \frac{2}{x} \times 1) (x^{2} - \frac{1}{x ^{2}}) = 0, x \neq = 0

Calculate

(2 x^{5} \times \frac{2}{x} \times 1) (x^{2} - \frac{1}{x ^{2}}) = 0

Multiply the terms

More Steps

Multiply the terms

2 x^{5} \times \frac{2}{x} \times 1

Rewrite the expression

2 x^{5} \times \frac{2}{x}

Cancel out the common factor x

2 x^{4} \times 2

Multiply the terms

4 x^{4}

4 x^{4} (x^{2} - \frac{1}{x ^{2}}) = 0

Subtract the terms

More Steps

Simplify

x^{2} - \frac{1}{x ^{2}}

Reduce fractions to a common denominator

\frac{x ^{2} \times x ^{2}}{x ^{2}} - \frac{1}{x ^{2}}

Write all numerators above the common denominator

\frac{x ^{2} \times x ^{2} - 1}{x ^{2}}

Multiply the terms

More Steps

Evaluate

x^{2} \times x^{2}

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

x^{2 + 2}

Add the numbers

x^{4}

\frac{x ^{4} - 1}{x ^{2}}

4 x^{4} \times \frac{x ^{4} - 1}{x ^{2}} = 0

Cancel out the common factor x^{2}

4 x^{2} (x^{4} - 1) = 0

Elimination the left coefficient

x^{2} (x^{4} - 1) = 0

Separate the equation into 2 possible cases

x^{2} = 0 x^{4} - 1 = 0

The only way a power can be 0 is when the base equals 0

x = 0 x^{4} - 1 = 0

Solve the equation

More Steps

Evaluate

x^{4} - 1 = 0

Move the constant to the right-hand side and change its sign

x^{4} = 0 + 1

Removing 0 doesn't change the value,so remove it from the expression

x^{4} = 1

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm 41

Simplify the expression

x = \pm 1

Separate the equation into 2 possible cases

x = 1 x = - 1

x = 0 x = 1 x = - 1

Check if the solution is in the defined range

x = 0 x = 1 x = - 1, x \neq = 0

Find the intersection of the solution and the defined range

x = 1 x = - 1

Solution

x_{1} = - 1, x_{2} = 1

Show Solution