Algebra Examples

$x^{4} - {(x - z)}^{4}$

Step 1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(x^{2})}^{2} - {(x - z)}^{4}$

Step 2

Rewrite ${(x - z)}^{4}$ as ${({(x - z)}^{2})}^{2}$ .

${(x^{2})}^{2} - {({(x - z)}^{2})}^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = {(x - z)}^{2}$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $x$ by $x$ .

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

Step 4.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.4

Multiply $z$ by $z$ by adding the exponents.

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Step 4.3.1.4.1

Move $z$ .

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

Step 4.3.1.4.2

Multiply $z$ by $z$ .

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

Step 4.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 4.3.1.6

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

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

Step 4.3.2

Subtract $z x$ from $- x z$ .

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Step 4.3.2.1

Move $z$ .

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

Step 4.3.2.2

Subtract $x z$ from $- x z$ .

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

Step 4.4

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

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

Step 4.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = x - z$ .

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

Step 4.6

Factor.

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Step 4.6.1

Simplify.

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Step 4.6.1.1

Remove unnecessary parentheses.

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

Step 4.6.1.2

Add $x$ and $x$ .

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

Step 4.6.1.3

Apply the distributive property.

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

Step 4.6.1.4

Multiply $- - z$ .

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Step 4.6.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.6.1.4.2

Multiply $z$ by $1$ .

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

Step 4.6.1.5

Subtract $x$ from $x$ .

$(2 x^{2} - 2 x z + z^{2}) ((2 x - z) (0 + z))$

Step 4.6.1.6

Add $0$ and $z$ .

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

Step 4.6.2

Remove unnecessary parentheses.

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