Solve (x^5)*(x-5)*(x^2-25)

Question

Simplify the expression

x^{8} - 25 x^{6} - 5 x^{7} + 125 x^{5}

Evaluate

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

Multiply the terms

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Evaluate

x^{5} (x - 5)

Apply the distributive property

x^{5} \times x - x^{5} \times 5

Multiply the terms

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Evaluate

x^{5} \times x

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

x^{5 + 1}

Add the numbers

x^{6}

x^{6} - x^{5} \times 5

Use the commutative property to reorder the terms

x^{6} - 5 x^{5}

(x^{6} - 5 x^{5}) (x^{2} - 25)

Apply the distributive property

x^{6} \times x^{2} - x^{6} \times 25 - 5 x^{5} \times x^{2} - (- 5 x^{5} \times 25)

Multiply the terms

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Evaluate

x^{6} \times x^{2}

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

x^{6 + 2}

Add the numbers

x^{8}

x^{8} - x^{6} \times 25 - 5 x^{5} \times x^{2} - (- 5 x^{5} \times 25)

Use the commutative property to reorder the terms

x^{8} - 25 x^{6} - 5 x^{5} \times x^{2} - (- 5 x^{5} \times 25)

Multiply the terms

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Evaluate

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

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

x^{5 + 2}

Add the numbers

x^{7}

x^{8} - 25 x^{6} - 5 x^{7} - (- 5 x^{5} \times 25)

Multiply the numbers

x^{8} - 25 x^{6} - 5 x^{7} - (- 125 x^{5})

Solution

x^{8} - 25 x^{6} - 5 x^{7} + 125 x^{5}

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Factor the expression

x^{5} (x - 5)^{2} (x + 5)

Evaluate

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

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

x^{5} (x - 5) (x - 5) (x + 5)

Solution

x^{5} (x - 5)^{2} (x + 5)

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Find the roots

x_{1} = - 5, x_{2} = 0, x_{3} = 5

Evaluate

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

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

(x^{5}) (x - 5) (x^{2} - 25) = 0

Calculate

x^{5} (x - 5) (x^{2} - 25) = 0

Separate the equation into 3 possible cases

x^{5} = 0 x - 5 = 0 x^{2} - 25 = 0

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

x = 0 x - 5 = 0 x^{2} - 25 = 0

Solve the equation

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Evaluate

x - 5 = 0

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

x = 0 + 5

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

x = 5

x = 0 x = 5 x^{2} - 25 = 0

Solve the equation

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Evaluate

x^{2} - 25 = 0

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

x^{2} = 0 + 25

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

x^{2} = 25

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

x = \pm 25

Simplify the expression

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Evaluate

25

Write the number in exponential form with the base of 5

5^{2}

Reduce the index of the radical and exponent with 2

5

x = \pm 5

Separate the equation into 2 possible cases

x = 5 x = - 5

x = 0 x = 5 x = 5 x = - 5

Find the union

x = 0 x = 5 x = - 5

Solution

x_{1} = - 5, x_{2} = 0, x_{3} = 5

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