Precalculus Examples

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

Step 1

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

${({(x + 2)}^{2})}^{2} - 16$

Step 2

Rewrite $16$ as $4^{2}$ .

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

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 $b = 4$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 4.3.1.2

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

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

Step 4.3.1.3

Multiply $2$ by $2$ .

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

Step 4.3.2

Add $2 x$ and $2 x$ .

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

Step 4.4

Add $4$ and $4$ .

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

Step 4.5

Rewrite $4$ as $2^{2}$ .

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

Step 4.6

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 $b = 2$ .

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

Step 4.7

Factor.

Tap for more steps...

Step 4.7.1

Simplify.

Tap for more steps...

Step 4.7.1.1

Add $2$ and $2$ .

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

Step 4.7.1.2

Subtract $2$ from $2$ .

$(x^{2} + 4 x + 8) ((x + 4) (x + 0))$

Step 4.7.1.3

Add $x$ and $0$ .

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

Step 4.7.2

Remove unnecessary parentheses.

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