Solve (x-1)(x-2)(x-3)(x-4)(x-5)

Question

Simplify the expression

x^{5} - 15 x^{4} + 85 x^{3} - 225 x^{2} + 274 x - 120

Evaluate

(x - 1) (x - 2) (x - 3) (x - 4) (x - 5)

Multiply the terms

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

Multiply the terms

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(x^{3} - 6 x^{2} + 11 x - 6) (x - 4) (x - 5)

Multiply the terms

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(x^{4} - 10 x^{3} + 35 x^{2} - 50 x + 24) (x - 5)

Apply the distributive property

x^{4} \times x - x^{4} \times 5 - 10 x^{3} \times x - (- 10 x^{3} \times 5) + 35 x^{2} \times x - 35 x^{2} \times 5 - 50 x \times x - (- 50 x \times 5) + 24 x - 24 \times 5

Multiply the terms

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x^{5} - x^{4} \times 5 - 10 x^{3} \times x - (- 10 x^{3} \times 5) + 35 x^{2} \times x - 35 x^{2} \times 5 - 50 x \times x - (- 50 x \times 5) + 24 x - 24 \times 5

Use the commutative property to reorder the terms

x^{5} - 5 x^{4} - 10 x^{3} \times x - (- 10 x^{3} \times 5) + 35 x^{2} \times x - 35 x^{2} \times 5 - 50 x \times x - (- 50 x \times 5) + 24 x - 24 \times 5

Multiply the terms

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x^{5} - 5 x^{4} - 10 x^{4} - (- 10 x^{3} \times 5) + 35 x^{2} \times x - 35 x^{2} \times 5 - 50 x \times x - (- 50 x \times 5) + 24 x - 24 \times 5

Multiply the numbers

x^{5} - 5 x^{4} - 10 x^{4} - (- 50 x^{3}) + 35 x^{2} \times x - 35 x^{2} \times 5 - 50 x \times x - (- 50 x \times 5) + 24 x - 24 \times 5

Multiply the terms

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x^{5} - 5 x^{4} - 10 x^{4} - (- 50 x^{3}) + 35 x^{3} - 35 x^{2} \times 5 - 50 x \times x - (- 50 x \times 5) + 24 x - 24 \times 5

Multiply the numbers

x^{5} - 5 x^{4} - 10 x^{4} - (- 50 x^{3}) + 35 x^{3} - 175 x^{2} - 50 x \times x - (- 50 x \times 5) + 24 x - 24 \times 5

Multiply the terms

x^{5} - 5 x^{4} - 10 x^{4} - (- 50 x^{3}) + 35 x^{3} - 175 x^{2} - 50 x^{2} - (- 50 x \times 5) + 24 x - 24 \times 5

Multiply the numbers

x^{5} - 5 x^{4} - 10 x^{4} - (- 50 x^{3}) + 35 x^{3} - 175 x^{2} - 50 x^{2} - (- 250 x) + 24 x - 24 \times 5

Multiply the numbers

x^{5} - 5 x^{4} - 10 x^{4} - (- 50 x^{3}) + 35 x^{3} - 175 x^{2} - 50 x^{2} - (- 250 x) + 24 x - 120

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^{5} - 5 x^{4} - 10 x^{4} + 50 x^{3} + 35 x^{3} - 175 x^{2} - 50 x^{2} + 250 x + 24 x - 120

Subtract the terms

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x^{5} - 15 x^{4} + 50 x^{3} + 35 x^{3} - 175 x^{2} - 50 x^{2} + 250 x + 24 x - 120

Add the terms

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x^{5} - 15 x^{4} + 85 x^{3} - 175 x^{2} - 50 x^{2} + 250 x + 24 x - 120

Subtract the terms

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x^{5} - 15 x^{4} + 85 x^{3} - 225 x^{2} + 250 x + 24 x - 120

Solution

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x^{5} - 15 x^{4} + 85 x^{3} - 225 x^{2} + 274 x - 120

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Find the roots

x_{1} = 1, x_{2} = 2, x_{3} = 3, x_{4} = 4, x_{5} = 5

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