Algebra Examples

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

Step 1

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

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

Step 2

Rewrite ${(y + 3)}^{4}$ as ${({(y + 3)}^{2})}^{2}$ .

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

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x - 2)}^{2}$ and $b = {(y + 3)}^{2}$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

Tap for more steps...

Step 4.3.1

Simplify each term.

Tap for more steps...

Step 4.3.1.1

Multiply $x$ by $x$ .

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

Step 4.3.1.2

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

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

Step 4.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 4.3.2

Subtract $2 x$ from $- 2 x$ .

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

Step 4.4

Rewrite ${(y + 3)}^{2}$ as $(y + 3) (y + 3)$ .

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

Step 4.5

Expand $(y + 3) (y + 3)$ using the FOIL Method.

Tap for more steps...

Step 4.5.1

Apply the distributive property.

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

Step 4.5.2

Apply the distributive property.

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

Step 4.5.3

Apply the distributive property.

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

Step 4.6

Simplify and combine like terms.

Tap for more steps...

Step 4.6.1

Simplify each term.

Tap for more steps...

Step 4.6.1.1

Multiply $y$ by $y$ .

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

Step 4.6.1.2

Move $3$ to the left of $y$ .

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

Step 4.6.1.3

Multiply $3$ by $3$ .

$(x^{2} - 4 x + 4 + y^{2} + 3 y + 3 y + 9) ({(x - 2)}^{2} - {(y + 3)}^{2})$

Step 4.6.2

Add $3 y$ and $3 y$ .

$(x^{2} - 4 x + 4 + y^{2} + 6 y + 9) ({(x - 2)}^{2} - {(y + 3)}^{2})$

Step 4.7

Add $4$ and $9$ .

$(x^{2} - 4 x + y^{2} + 6 y + 13) ({(x - 2)}^{2} - {(y + 3)}^{2})$

Step 4.8

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 $b = y + 3$ .

$(x^{2} - 4 x + y^{2} + 6 y + 13) ((x - 2 + y + 3) (x - 2 - (y + 3)))$

Step 4.9

Factor.

Tap for more steps...

Step 4.9.1

Simplify.

Tap for more steps...

Step 4.9.1.1

Remove unnecessary parentheses.

$(x^{2} - 4 x + y^{2} + 6 y + 13) ((x - 2 + y + 3) (x - 2 - (y + 3)))$

Step 4.9.1.2

Add $- 2$ and $3$ .

$(x^{2} - 4 x + y^{2} + 6 y + 13) ((x + y + 1) (x - 2 - (y + 3)))$

Step 4.9.1.3

Apply the distributive property.

$(x^{2} - 4 x + y^{2} + 6 y + 13) ((x + y + 1) (x - 2 - y - 1 \cdot 3))$

Step 4.9.1.4

Multiply $- 1$ by $3$ .

$(x^{2} - 4 x + y^{2} + 6 y + 13) ((x + y + 1) (x - 2 - y - 3))$

Step 4.9.1.5

Subtract $3$ from $- 2$ .

$(x^{2} - 4 x + y^{2} + 6 y + 13) ((x + y + 1) (x - y - 5))$

Step 4.9.2

Remove unnecessary parentheses.

$(x^{2} - 4 x + y^{2} + 6 y + 13) (x + y + 1) (x - y - 5)$