Algebra Examples

$(x - 4) (2 x - 1) (2 x + 1) (x - 1)$

Step 1

Expand $(x - 4) (2 x - 1)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$(x (2 x - 1) - 4 (2 x - 1)) (2 x + 1) (x - 1)$

Step 1.2

Apply the distributive property.

$(x (2 x) + x \cdot - 1 - 4 (2 x - 1)) (2 x + 1) (x - 1)$

Step 1.3

Apply the distributive property.

$(x (2 x) + x \cdot - 1 - 4 (2 x) - 4 \cdot - 1) (2 x + 1) (x - 1)$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$(2 x \cdot x + x \cdot - 1 - 4 (2 x) - 4 \cdot - 1) (2 x + 1) (x - 1)$

Step 2.1.2

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

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Step 2.1.2.1

Move $x$ .

$(2 (x \cdot x) + x \cdot - 1 - 4 (2 x) - 4 \cdot - 1) (2 x + 1) (x - 1)$

Step 2.1.2.2

Multiply $x$ by $x$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Multiply $2$ by $- 4$ .

$(2 x^{2} - x - 8 x - 4 \cdot - 1) (2 x + 1) (x - 1)$

Step 2.1.6

Multiply $- 4$ by $- 1$ .

$(2 x^{2} - x - 8 x + 4) (2 x + 1) (x - 1)$

Step 2.2

Subtract $8 x$ from $- x$ .

$(2 x^{2} - 9 x + 4) (2 x + 1) (x - 1)$

Step 3

Expand $(2 x^{2} - 9 x + 4) (2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$(2 x^{2} (2 x) + 2 x^{2} \cdot 1 - 9 x (2 x) - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

$(2 \cdot 2 x^{2} x + 2 x^{2} \cdot 1 - 9 x (2 x) - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.2

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

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Step 4.1.2.1

Move $x$ .

$(2 \cdot 2 (x \cdot x^{2}) + 2 x^{2} \cdot 1 - 9 x (2 x) - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.2.2

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

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Step 4.1.2.2.1

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

$(2 \cdot 2 (x^{1} x^{2}) + 2 x^{2} \cdot 1 - 9 x (2 x) - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.2.2.2

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

$(2 \cdot 2 x^{1 + 2} + 2 x^{2} \cdot 1 - 9 x (2 x) - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.2.3

Add $1$ and $2$ .

$(2 \cdot 2 x^{3} + 2 x^{2} \cdot 1 - 9 x (2 x) - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.3

Multiply $2$ by $2$ .

$(4 x^{3} + 2 x^{2} \cdot 1 - 9 x (2 x) - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.4

Multiply $2$ by $1$ .

$(4 x^{3} + 2 x^{2} - 9 x (2 x) - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.5

Rewrite using the commutative property of multiplication.

$(4 x^{3} + 2 x^{2} - 9 \cdot 2 x \cdot x - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.6

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

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Step 4.1.6.1

Move $x$ .

$(4 x^{3} + 2 x^{2} - 9 \cdot 2 (x \cdot x) - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.6.2

Multiply $x$ by $x$ .

$(4 x^{3} + 2 x^{2} - 9 \cdot 2 x^{2} - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.7

Multiply $- 9$ by $2$ .

$(4 x^{3} + 2 x^{2} - 18 x^{2} - 9 x \cdot 1 + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.8

Multiply $- 9$ by $1$ .

$(4 x^{3} + 2 x^{2} - 18 x^{2} - 9 x + 4 (2 x) + 4 \cdot 1) (x - 1)$

Step 4.1.9

Multiply $2$ by $4$ .

$(4 x^{3} + 2 x^{2} - 18 x^{2} - 9 x + 8 x + 4 \cdot 1) (x - 1)$

Step 4.1.10

Multiply $4$ by $1$ .

$(4 x^{3} + 2 x^{2} - 18 x^{2} - 9 x + 8 x + 4) (x - 1)$

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Subtract $18 x^{2}$ from $2 x^{2}$ .

$(4 x^{3} - 16 x^{2} - 9 x + 8 x + 4) (x - 1)$

Step 4.2.2

Add $- 9 x$ and $8 x$ .

$(4 x^{3} - 16 x^{2} - x + 4) (x - 1)$

Step 5

Expand $(4 x^{3} - 16 x^{2} - x + 4) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$4 x^{3} x + 4 x^{3} \cdot - 1 - 16 x^{2} x - 16 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6

Simplify terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

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Step 6.1.1.1

Move $x$ .

$4 (x \cdot x^{3}) + 4 x^{3} \cdot - 1 - 16 x^{2} x - 16 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.1.2

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

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Step 6.1.1.2.1

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

$4 (x^{1} x^{3}) + 4 x^{3} \cdot - 1 - 16 x^{2} x - 16 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.1.2.2

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

$4 x^{1 + 3} + 4 x^{3} \cdot - 1 - 16 x^{2} x - 16 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.1.3

Add $1$ and $3$ .

$4 x^{4} + 4 x^{3} \cdot - 1 - 16 x^{2} x - 16 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.2

Multiply $- 1$ by $4$ .

$4 x^{4} - 4 x^{3} - 16 x^{2} x - 16 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.3

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

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Step 6.1.3.1

Move $x$ .

$4 x^{4} - 4 x^{3} - 16 (x \cdot x^{2}) - 16 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.3.2

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

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Step 6.1.3.2.1

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

$4 x^{4} - 4 x^{3} - 16 (x^{1} x^{2}) - 16 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.3.2.2

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

$4 x^{4} - 4 x^{3} - 16 x^{1 + 2} - 16 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.3.3

Add $1$ and $2$ .

$4 x^{4} - 4 x^{3} - 16 x^{3} - 16 x^{2} \cdot - 1 - x \cdot x - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.4

Multiply $- 1$ by $- 16$ .

$4 x^{4} - 4 x^{3} - 16 x^{3} + 16 x^{2} - x \cdot x - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.5

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

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Step 6.1.5.1

Move $x$ .

$4 x^{4} - 4 x^{3} - 16 x^{3} + 16 x^{2} - (x \cdot x) - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.5.2

Multiply $x$ by $x$ .

$4 x^{4} - 4 x^{3} - 16 x^{3} + 16 x^{2} - x^{2} - x \cdot - 1 + 4 x + 4 \cdot - 1$

Step 6.1.6

Multiply $- x \cdot - 1$ .

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Step 6.1.6.1

Multiply $- 1$ by $- 1$ .

$4 x^{4} - 4 x^{3} - 16 x^{3} + 16 x^{2} - x^{2} + 1 x + 4 x + 4 \cdot - 1$

Step 6.1.6.2

Multiply $x$ by $1$ .

$4 x^{4} - 4 x^{3} - 16 x^{3} + 16 x^{2} - x^{2} + x + 4 x + 4 \cdot - 1$

Step 6.1.7

Multiply $4$ by $- 1$ .

$4 x^{4} - 4 x^{3} - 16 x^{3} + 16 x^{2} - x^{2} + x + 4 x - 4$

Step 6.2

Simplify by adding terms.

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Step 6.2.1

Subtract $16 x^{3}$ from $- 4 x^{3}$ .

$4 x^{4} - 20 x^{3} + 16 x^{2} - x^{2} + x + 4 x - 4$

Step 6.2.2

Subtract $x^{2}$ from $16 x^{2}$ .

$4 x^{4} - 20 x^{3} + 15 x^{2} + x + 4 x - 4$

Step 6.2.3

Add $x$ and $4 x$ .

$4 x^{4} - 20 x^{3} + 15 x^{2} + 5 x - 4$