Solve (x-2)*(x-1)^2*(x 1)^3*(x-2)

Question

Simplify the expression

x^{7} - 6 x^{6} + 13 x^{5} - 12 x^{4} + 4 x^{3}

Evaluate

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

Any expression multiplied by 1 remains the same

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

Use the commutative property to reorder the terms

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

Multiply the terms

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

Expand the expression

More Steps

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

Expand the expression

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

Multiply the terms

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(x^{5} - 4 x^{4} + 4 x^{3}) (x^{2} - 2 x + 1)

Apply the distributive property

x^{5} \times x^{2} - x^{5} \times 2 x + x^{5} \times 1 - 4 x^{4} \times x^{2} - (- 4 x^{4} \times 2 x) - 4 x^{4} \times 1 + 4 x^{3} \times x^{2} - 4 x^{3} \times 2 x + 4 x^{3} \times 1

Multiply the terms

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x^{7} - x^{5} \times 2 x + x^{5} \times 1 - 4 x^{4} \times x^{2} - (- 4 x^{4} \times 2 x) - 4 x^{4} \times 1 + 4 x^{3} \times x^{2} - 4 x^{3} \times 2 x + 4 x^{3} \times 1

Multiply the terms

More Steps

x^{7} - 2 x^{6} + x^{5} \times 1 - 4 x^{4} \times x^{2} - (- 4 x^{4} \times 2 x) - 4 x^{4} \times 1 + 4 x^{3} \times x^{2} - 4 x^{3} \times 2 x + 4 x^{3} \times 1

Any expression multiplied by 1 remains the same

x^{7} - 2 x^{6} + x^{5} - 4 x^{4} \times x^{2} - (- 4 x^{4} \times 2 x) - 4 x^{4} \times 1 + 4 x^{3} \times x^{2} - 4 x^{3} \times 2 x + 4 x^{3} \times 1

Multiply the terms

More Steps

x^{7} - 2 x^{6} + x^{5} - 4 x^{6} - (- 4 x^{4} \times 2 x) - 4 x^{4} \times 1 + 4 x^{3} \times x^{2} - 4 x^{3} \times 2 x + 4 x^{3} \times 1

Multiply the terms

More Steps

x^{7} - 2 x^{6} + x^{5} - 4 x^{6} - (- 8 x^{5}) - 4 x^{4} \times 1 + 4 x^{3} \times x^{2} - 4 x^{3} \times 2 x + 4 x^{3} \times 1

Any expression multiplied by 1 remains the same

x^{7} - 2 x^{6} + x^{5} - 4 x^{6} - (- 8 x^{5}) - 4 x^{4} + 4 x^{3} \times x^{2} - 4 x^{3} \times 2 x + 4 x^{3} \times 1

Multiply the terms

More Steps

x^{7} - 2 x^{6} + x^{5} - 4 x^{6} - (- 8 x^{5}) - 4 x^{4} + 4 x^{5} - 4 x^{3} \times 2 x + 4 x^{3} \times 1

Multiply the terms

More Steps

x^{7} - 2 x^{6} + x^{5} - 4 x^{6} - (- 8 x^{5}) - 4 x^{4} + 4 x^{5} - 8 x^{4} + 4 x^{3} \times 1

Any expression multiplied by 1 remains the same

x^{7} - 2 x^{6} + x^{5} - 4 x^{6} - (- 8 x^{5}) - 4 x^{4} + 4 x^{5} - 8 x^{4} + 4 x^{3}

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^{7} - 2 x^{6} + x^{5} - 4 x^{6} + 8 x^{5} - 4 x^{4} + 4 x^{5} - 8 x^{4} + 4 x^{3}

Subtract the terms

More Steps

x^{7} - 6 x^{6} + x^{5} + 8 x^{5} - 4 x^{4} + 4 x^{5} - 8 x^{4} + 4 x^{3}

Add the terms

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x^{7} - 6 x^{6} + 13 x^{5} - 4 x^{4} - 8 x^{4} + 4 x^{3}

Solution

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x^{7} - 6 x^{6} + 13 x^{5} - 12 x^{4} + 4 x^{3}

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Find the roots

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

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