Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 3.2

Subtract $x$ from $- x$ .

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

Step 4

Apply the distributive property.

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

Step 5

Simplify.

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Step 5.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 5.1.1

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

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

Step 5.1.2

Add $3$ and $2$ .

$(x^{5} + x^{3} (- 2 x) + x^{3} \cdot 1) (2 + x) (2 - x)$

Step 5.2

Rewrite using the commutative property of multiplication.

$(x^{5} - 2 x^{3} x + x^{3} \cdot 1) (2 + x) (2 - x)$

Step 5.3

Multiply $x^{3}$ by $1$ .

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

Step 6

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

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Step 6.1

Move $x$ .

$(x^{5} - 2 (x \cdot x^{3}) + x^{3}) (2 + x) (2 - x)$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.3

Add $1$ and $3$ .

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

Step 7

Expand $(x^{5} - 2 x^{4} + x^{3}) (2 + x)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 8

Simplify terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Move $2$ to the left of $x^{5}$ .

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

Step 8.1.2

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 8.1.2.1

Multiply $x^{5}$ by $x$ .

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Step 8.1.2.1.1

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

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

Step 8.1.2.1.2

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

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

Step 8.1.2.2

Add $5$ and $1$ .

$(2 x^{5} + x^{6} - 2 x^{4} \cdot 2 - 2 x^{4} x + x^{3} \cdot 2 + x^{3} x) (2 - x)$

Step 8.1.3

Multiply $2$ by $- 2$ .

$(2 x^{5} + x^{6} - 4 x^{4} - 2 x^{4} x + x^{3} \cdot 2 + x^{3} x) (2 - x)$

Step 8.1.4

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 8.1.4.1

Move $x$ .

$(2 x^{5} + x^{6} - 4 x^{4} - 2 (x \cdot x^{4}) + x^{3} \cdot 2 + x^{3} x) (2 - x)$

Step 8.1.4.2

Multiply $x$ by $x^{4}$ .

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Step 8.1.4.2.1

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

$(2 x^{5} + x^{6} - 4 x^{4} - 2 (x^{1} x^{4}) + x^{3} \cdot 2 + x^{3} x) (2 - x)$

Step 8.1.4.2.2

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

$(2 x^{5} + x^{6} - 4 x^{4} - 2 x^{1 + 4} + x^{3} \cdot 2 + x^{3} x) (2 - x)$

Step 8.1.4.3

Add $1$ and $4$ .

$(2 x^{5} + x^{6} - 4 x^{4} - 2 x^{5} + x^{3} \cdot 2 + x^{3} x) (2 - x)$

Step 8.1.5

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

$(2 x^{5} + x^{6} - 4 x^{4} - 2 x^{5} + 2 \cdot x^{3} + x^{3} x) (2 - x)$

Step 8.1.6

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

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Step 8.1.6.1

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

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Step 8.1.6.1.1

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

$(2 x^{5} + x^{6} - 4 x^{4} - 2 x^{5} + 2 x^{3} + x^{3} x^{1}) (2 - x)$

Step 8.1.6.1.2

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

$(2 x^{5} + x^{6} - 4 x^{4} - 2 x^{5} + 2 x^{3} + x^{3 + 1}) (2 - x)$

Step 8.1.6.2

Add $3$ and $1$ .

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

Step 8.2

Simplify by adding terms.

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Step 8.2.1

Combine the opposite terms in $2 x^{5} + x^{6} - 4 x^{4} - 2 x^{5} + 2 x^{3} + x^{4}$ .

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Step 8.2.1.1

Subtract $2 x^{5}$ from $2 x^{5}$ .

$(x^{6} - 4 x^{4} + 0 + 2 x^{3} + x^{4}) (2 - x)$

Step 8.2.1.2

Add $x^{6} - 4 x^{4}$ and $0$ .

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

Step 8.2.2

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

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

Step 9

Expand $(x^{6} - 3 x^{4} + 2 x^{3}) (2 - x)$ by multiplying each term in the first expression by each term in the second expression.

$x^{6} \cdot 2 + x^{6} (- x) - 3 x^{4} \cdot 2 - 3 x^{4} (- x) + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10

Simplify terms.

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Step 10.1

Simplify each term.

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Step 10.1.1

Move $2$ to the left of $x^{6}$ .

$2 \cdot x^{6} + x^{6} (- x) - 3 x^{4} \cdot 2 - 3 x^{4} (- x) + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.2

Rewrite using the commutative property of multiplication.

$2 x^{6} - x^{6} x - 3 x^{4} \cdot 2 - 3 x^{4} (- x) + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.3

Multiply $x^{6}$ by $x$ by adding the exponents.

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Step 10.1.3.1

Move $x$ .

$2 x^{6} - (x \cdot x^{6}) - 3 x^{4} \cdot 2 - 3 x^{4} (- x) + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.3.2

Multiply $x$ by $x^{6}$ .

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Step 10.1.3.2.1

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

$2 x^{6} - (x^{1} x^{6}) - 3 x^{4} \cdot 2 - 3 x^{4} (- x) + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.3.2.2

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

$2 x^{6} - x^{1 + 6} - 3 x^{4} \cdot 2 - 3 x^{4} (- x) + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.3.3

Add $1$ and $6$ .

$2 x^{6} - x^{7} - 3 x^{4} \cdot 2 - 3 x^{4} (- x) + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.4

Multiply $2$ by $- 3$ .

$2 x^{6} - x^{7} - 6 x^{4} - 3 x^{4} (- x) + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.5

Rewrite using the commutative property of multiplication.

$2 x^{6} - x^{7} - 6 x^{4} - 3 \cdot - 1 x^{4} x + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.6

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 10.1.6.1

Move $x$ .

$2 x^{6} - x^{7} - 6 x^{4} - 3 \cdot - 1 (x \cdot x^{4}) + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.6.2

Multiply $x$ by $x^{4}$ .

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Step 10.1.6.2.1

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

$2 x^{6} - x^{7} - 6 x^{4} - 3 \cdot - 1 (x^{1} x^{4}) + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.6.2.2

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

$2 x^{6} - x^{7} - 6 x^{4} - 3 \cdot - 1 x^{1 + 4} + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.6.3

Add $1$ and $4$ .

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

Step 10.1.7

Multiply $- 3$ by $- 1$ .

$2 x^{6} - x^{7} - 6 x^{4} + 3 x^{5} + 2 x^{3} \cdot 2 + 2 x^{3} (- x)$

Step 10.1.8

Multiply $2$ by $2$ .

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

Step 10.1.9

Rewrite using the commutative property of multiplication.

$2 x^{6} - x^{7} - 6 x^{4} + 3 x^{5} + 4 x^{3} + 2 \cdot - 1 x^{3} x$

Step 10.1.10

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

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Step 10.1.10.1

Move $x$ .

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

Step 10.1.10.2

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

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Step 10.1.10.2.1

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

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

Step 10.1.10.2.2

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

$2 x^{6} - x^{7} - 6 x^{4} + 3 x^{5} + 4 x^{3} + 2 \cdot - 1 x^{1 + 3}$

Step 10.1.10.3

Add $1$ and $3$ .

$2 x^{6} - x^{7} - 6 x^{4} + 3 x^{5} + 4 x^{3} + 2 \cdot - 1 x^{4}$

Step 10.1.11

Multiply $2$ by $- 1$ .

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

Step 10.2

Simplify by adding terms.

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Step 10.2.1

Subtract $2 x^{4}$ from $- 6 x^{4}$ .

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

Step 10.2.2

Simplify the expression.

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Step 10.2.2.1

Move $- 8 x^{4}$ .

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

Step 10.2.2.2

Reorder $2 x^{6}$ and $- x^{7}$ .

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