Algebra Examples

$x {(x - 1)}^{2} (x + 2) = 0$

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$x ((x - 1) (x - 1)) (x + 2)$

Step 1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$x (x (x - 1) - 1 (x - 1)) (x + 2)$

Step 1.2.2

Apply the distributive property.

$x (x \cdot x + x \cdot - 1 - 1 (x - 1)) (x + 2)$

Step 1.2.3

Apply the distributive property.

$x (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) (x + 2)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$x (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) (x + 2)$

Step 1.3.1.2

Move $- 1$ to the left of $x$ .

$x (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) (x + 2)$

Step 1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$x (x^{2} - x - 1 x - 1 \cdot - 1) (x + 2)$

Step 1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$x (x^{2} - x - x - 1 \cdot - 1) (x + 2)$

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.3.2

Subtract $x$ from $- x$ .

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

Step 1.4

Apply the distributive property.

$(x \cdot x^{2} + x (- 2 x) + x \cdot 1) (x + 2)$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.5.1.1

Multiply $x$ by $x^{2}$ .

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Step 1.5.1.1.1

Raise $x$ to the power of $1$ .

$(x^{1} x^{2} + x (- 2 x) + x \cdot 1) (x + 2)$

Step 1.5.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{1 + 2} + x (- 2 x) + x \cdot 1) (x + 2)$

Step 1.5.1.2

Add $1$ and $2$ .

$(x^{3} + x (- 2 x) + x \cdot 1) (x + 2)$

Step 1.5.2

Rewrite using the commutative property of multiplication.

$(x^{3} - 2 x \cdot x + x \cdot 1) (x + 2)$

Step 1.5.3

Multiply $x$ by $1$ .

$(x^{3} - 2 x \cdot x + x) (x + 2)$

Step 1.6

Multiply $x$ by $x$ by adding the exponents.

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Step 1.6.1

Move $x$ .

$(x^{3} - 2 (x \cdot x) + x) (x + 2)$

Step 1.6.2

Multiply $x$ by $x$ .

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

Step 1.7

Expand $(x^{3} - 2 x^{2} + x) (x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x + x^{3} \cdot 2 - 2 x^{2} x - 2 x^{2} \cdot 2 + x \cdot x + x \cdot 2$

Step 1.8

Simplify terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.8.1.1.1

Multiply $x^{3}$ by $x$ .

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Step 1.8.1.1.1.1

Raise $x$ to the power of $1$ .

$x^{3} x^{1} + x^{3} \cdot 2 - 2 x^{2} x - 2 x^{2} \cdot 2 + x \cdot x + x \cdot 2$

Step 1.8.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3 + 1} + x^{3} \cdot 2 - 2 x^{2} x - 2 x^{2} \cdot 2 + x \cdot x + x \cdot 2$

Step 1.8.1.1.2

Add $3$ and $1$ .

$x^{4} + x^{3} \cdot 2 - 2 x^{2} x - 2 x^{2} \cdot 2 + x \cdot x + x \cdot 2$

Step 1.8.1.2

Move $2$ to the left of $x^{3}$ .

$x^{4} + 2 \cdot x^{3} - 2 x^{2} x - 2 x^{2} \cdot 2 + x \cdot x + x \cdot 2$

Step 1.8.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.8.1.3.1

Move $x$ .

$x^{4} + 2 x^{3} - 2 (x \cdot x^{2}) - 2 x^{2} \cdot 2 + x \cdot x + x \cdot 2$

Step 1.8.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.8.1.3.2.1

Raise $x$ to the power of $1$ .

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

Step 1.8.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4} + 2 x^{3} - 2 x^{1 + 2} - 2 x^{2} \cdot 2 + x \cdot x + x \cdot 2$

Step 1.8.1.3.3

Add $1$ and $2$ .

$x^{4} + 2 x^{3} - 2 x^{3} - 2 x^{2} \cdot 2 + x \cdot x + x \cdot 2$

Step 1.8.1.4

Multiply $2$ by $- 2$ .

$x^{4} + 2 x^{3} - 2 x^{3} - 4 x^{2} + x \cdot x + x \cdot 2$

Step 1.8.1.5

Multiply $x$ by $x$ .

$x^{4} + 2 x^{3} - 2 x^{3} - 4 x^{2} + x^{2} + x \cdot 2$

Step 1.8.1.6

Move $2$ to the left of $x$ .

$x^{4} + 2 x^{3} - 2 x^{3} - 4 x^{2} + x^{2} + 2 x$

Step 1.8.2

Simplify by adding terms.

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Step 1.8.2.1

Combine the opposite terms in $x^{4} + 2 x^{3} - 2 x^{3} - 4 x^{2} + x^{2} + 2 x$ .

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Step 1.8.2.1.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$x^{4} + 0 - 4 x^{2} + x^{2} + 2 x$

Step 1.8.2.1.2

Add $x^{4}$ and $0$ .

$x^{4} - 4 x^{2} + x^{2} + 2 x$

Step 1.8.2.2

Add $- 4 x^{2}$ and $x^{2}$ .

$x^{4} - 3 x^{2} + 2 x$

Step 2

The largest exponent is the degree of the polynomial.

$4$