Solve x^3 x (x-6)^3

Question

Simplify the expression

x^{7} - 18 x^{6} + 108 x^{5} - 216 x^{4}

Evaluate

x^{3} \times x (x - 6)^{3}

Multiply the terms with the same base by adding their exponents

x^{3 + 1} (x - 6)^{3}

Add the numbers

x^{4} (x - 6)^{3}

Expand the expression

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Evaluate

(x - 6)^{3}

Use (a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3} to expand the expression

x^{3} - 3 x^{2} \times 6 + 3 x \times 6^{2} - 6^{3}

Calculate

x^{3} - 18 x^{2} + 108 x - 216

x^{4} (x^{3} - 18 x^{2} + 108 x - 216)

Apply the distributive property

x^{4} \times x^{3} - x^{4} \times 18 x^{2} + x^{4} \times 108 x - x^{4} \times 216

Multiply the terms

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Evaluate

x^{4} \times x^{3}

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

x^{4 + 3}

Add the numbers

x^{7}

x^{7} - x^{4} \times 18 x^{2} + x^{4} \times 108 x - x^{4} \times 216

Multiply the terms

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Evaluate

x^{4} \times 18 x^{2}

Use the commutative property to reorder the terms

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

Multiply the terms

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Evaluate

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

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

x^{4 + 2}

Add the numbers

x^{6}

18 x^{6}

x^{7} - 18 x^{6} + x^{4} \times 108 x - x^{4} \times 216

Multiply the terms

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Evaluate

x^{4} \times 108 x

Use the commutative property to reorder the terms

108 x^{4} \times x

Multiply the terms

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Evaluate

x^{4} \times x

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

x^{4 + 1}

Add the numbers

x^{5}

108 x^{5}

x^{7} - 18 x^{6} + 108 x^{5} - x^{4} \times 216

Solution

x^{7} - 18 x^{6} + 108 x^{5} - 216 x^{4}

Show Solution

Find the roots

x_{1} = 0, x_{2} = 6

Evaluate

x^{3} \times x (x - 6)^{3}

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

x^{3} \times x (x - 6)^{3} = 0

Multiply

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Multiply the terms

x^{3} \times x (x - 6)^{3}

Multiply the terms with the same base by adding their exponents

x^{3 + 1} (x - 6)^{3}

Add the numbers

x^{4} (x - 6)^{3}

x^{4} (x - 6)^{3} = 0

Separate the equation into 2 possible cases

x^{4} = 0 (x - 6)^{3} = 0

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

x = 0 (x - 6)^{3} = 0

Solve the equation

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Evaluate

(x - 6)^{3} = 0

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

x - 6 = 0

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

x = 0 + 6

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

x = 6

x = 0 x = 6

Solution

x_{1} = 0, x_{2} = 6

Show Solution