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

Question

Simplify the expression

x^{6} - 3 x^{5} + 2 x^{4}

Evaluate

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

Remove the parentheses

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

Rewrite the expression

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

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

Multiply the terms

More Steps

Evaluate

x^{4} (x - 1)

Apply the distributive property

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

Multiply the terms

More Steps

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}

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

Any expression multiplied by 1 remains the same

x^{5} - x^{4}

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

Apply the distributive property

x^{5} \times x - x^{5} \times 2 - x^{4} \times x - (- x^{4} \times 2)

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

Use the commutative property to reorder the terms

x^{6} - 2 x^{5} - x^{4} \times x - (- x^{4} \times 2)

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}

x^{6} - 2 x^{5} - x^{5} - (- x^{4} \times 2)

Use the commutative property to reorder the terms

x^{6} - 2 x^{5} - x^{5} - (- 2 x^{4})

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

x^{6} - 2 x^{5} - x^{5} + 2 x^{4}

Solution

More Steps

Evaluate

- 2 x^{5} - x^{5}

Collect like terms by calculating the sum or difference of their coefficients

(- 2 - 1) x^{5}

Subtract the numbers

- 3 x^{5}

x^{6} - 3 x^{5} + 2 x^{4}

Show Solution

Find the roots

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

Evaluate

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

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

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

Any expression multiplied by 1 remains the same

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

Calculate

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

Multiply

More Steps

Multiply the terms

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

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

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

Separate the equation into 3 possible cases

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

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

x = 0 x - 1 = 0 x - 2 = 0

Solve the equation

More Steps

Evaluate

x - 1 = 0

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

x = 0 + 1

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

x = 1

x = 0 x = 1 x - 2 = 0

Solve the equation

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Evaluate

x - 2 = 0

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

x = 0 + 2

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

x = 2

x = 0 x = 1 x = 2

Solution

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

Show Solution