Algebra Examples

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

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

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

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $x$ by $x$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Subtract $x$ from $- x$ .

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

Step 5

Expand $(x^{2} - 2 x + 1) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 6

Simplify terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

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Step 6.1.1.1

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

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Step 6.1.1.1.1

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

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

Step 6.1.1.1.2

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

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

Step 6.1.1.2

Add $2$ and $1$ .

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

Step 6.1.2

Multiply $x^{2}$ by $1$ .

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

Step 6.1.3

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

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Step 6.1.3.1

Move $x$ .

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

Step 6.1.3.2

Multiply $x$ by $x$ .

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

Step 6.1.4

Multiply $- 2$ by $1$ .

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

Step 6.1.5

Multiply $x$ by $1$ .

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

Step 6.1.6

Multiply $1$ by $1$ .

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

Step 6.2

Simplify by adding terms.

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Step 6.2.1

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

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

Step 6.2.2

Add $- 2 x$ and $x$ .

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