Solve (x^3-1)(4x-25 x^2)

Question

Simplify the expression

4 x^{4} - 25 x^{5} - 4 x + 25 x^{2}

Evaluate

(x^{3} - 1) (4 x - 25 x^{2})

Apply the distributive property

x^{3} \times 4 x - x^{3} \times 25 x^{2} - 4 x - (- 25 x^{2})

Multiply the terms

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Evaluate

x^{3} \times 4 x

Use the commutative property to reorder the terms

4 x^{3} \times x

Multiply the terms

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Evaluate

x^{3} \times x

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

x^{3 + 1}

Add the numbers

x^{4}

4 x^{4}

4 x^{4} - x^{3} \times 25 x^{2} - 4 x - (- 25 x^{2})

Multiply the terms

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Evaluate

x^{3} \times 25 x^{2}

Use the commutative property to reorder the terms

25 x^{3} \times x^{2}

Multiply the terms

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Evaluate

x^{3} \times x^{2}

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

x^{3 + 2}

Add the numbers

x^{5}

25 x^{5}

4 x^{4} - 25 x^{5} - 4 x - (- 25 x^{2})

Solution

4 x^{4} - 25 x^{5} - 4 x + 25 x^{2}

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Factor the expression

x (x - 1) (x^{2} + x + 1) (4 - 25 x)

Evaluate

(x^{3} - 1) (4 x - 25 x^{2})

Factor the expression

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Evaluate

x^{3} - 1

Calculate

x^{3} + x^{2} + x - x^{2} - x - 1

Rewrite the expression

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

Factor out x from the expression

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

Factor out - 1 from the expression

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

Factor out x^{2} + x + 1 from the expression

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

(x - 1) (x^{2} + x + 1) (4 x - 25 x^{2})

Factor the expression

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Evaluate

4 x - 25 x^{2}

Rewrite the expression

x \times 4 - x \times 25 x

Factor out x from the expression

x (4 - 25 x)

(x - 1) (x^{2} + x + 1) x (4 - 25 x)

Solution

x (x - 1) (x^{2} + x + 1) (4 - 25 x)

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Find the roots

x_{1} = 0, x_{2} = \frac{4}{25}, x_{3} = 1

Alternative Form

x_{1} = 0, x_{2} = 0.16, x_{3} = 1

Evaluate

(x^{3} - 1) (4 x - 25 x^{2})

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

(x^{3} - 1) (4 x - 25 x^{2}) = 0

Separate the equation into 2 possible cases

x^{3} - 1 = 0 4 x - 25 x^{2} = 0

Solve the equation

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Evaluate

x^{3} - 1 = 0

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

x^{3} = 0 + 1

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

x^{3} = 1

Take the 3 -th root on both sides of the equation

3 x^{3} = 31

Calculate

x = 31

Simplify the root

x = 1

x = 1 4 x - 25 x^{2} = 0

Solve the equation

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Evaluate

4 x - 25 x^{2} = 0

Factor the expression

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Evaluate

4 x - 25 x^{2}

Rewrite the expression

x \times 4 - x \times 25 x

Factor out x from the expression

x (4 - 25 x)

x (4 - 25 x) = 0

When the product of factors equals 0,at least one factor is 0

x = 0 4 - 25 x = 0

Solve the equation for x

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Evaluate

4 - 25 x = 0

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

- 25 x = 0 - 4

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

- 25 x = - 4

Change the signs on both sides of the equation

25 x = 4

Divide both sides

\frac{25 x}{25} = \frac{4}{25}

Divide the numbers

x = \frac{4}{25}

x = 0 x = \frac{4}{25}

x = 1 x = 0 x = \frac{4}{25}

Solution

x_{1} = 0, x_{2} = \frac{4}{25}, x_{3} = 1

Alternative Form

x_{1} = 0, x_{2} = 0.16, x_{3} = 1

Show Solution