Algebra Examples

$(y^{4} + 8 y^{3} + 16 y^{2}) - (y^{2} + 8 y + 16)$

Step 1

Remove parentheses.

$y^{4} + 8 y^{3} + 16 y^{2} - (y^{2} + 8 y + 16)$

Step 2

Factor using the perfect square rule.

Tap for more steps...

Step 2.1

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

${(y^{2})}^{2} + 8 y^{3} + 16 y^{2} - (y^{2} + 8 y + 16)$

Step 2.2

Rewrite $16 y^{2}$ as ${(4 y)}^{2}$ .

${(y^{2})}^{2} + 8 y^{3} + {(4 y)}^{2} - (y^{2} + 8 y + 16)$

Step 2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 y^{3} = 2 \cdot y^{2} \cdot (4 y)$

Step 2.4

Rewrite the polynomial.

${(y^{2})}^{2} + 2 \cdot y^{2} \cdot (4 y) + {(4 y)}^{2} - (y^{2} + 8 y + 16)$

Step 2.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y^{2}$ and $b = 4 y$ .

${(y^{2} + 4 y)}^{2} - (y^{2} + 8 y + 16)$

Step 3

Rewrite $y^{2} + 8 y + 16$ as ${(y + 4)}^{2}$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 y = 2 \cdot y \cdot 4$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 4$ .

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

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y^{2} + 4 y$ and $b = y + 4$ .

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Factor out the greatest common factor from each group.

Tap for more steps...

Step 5.1.1

Group the first two terms and the last two terms.

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

Step 5.1.2

Factor out the greatest common factor (GCF) from each group.

$(y (y + 4) + 1 (y + 4)) (y^{2} + 4 y - (y + 4))$

Step 5.2

Factor the polynomial by factoring out the greatest common factor, $y + 4$ .

$(y + 4) (y + 1) (y^{2} + 4 y - (y + 4))$

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $- 1$ by $4$ .

$(y + 4) (y + 1) (y^{2} + 4 y - y - 4)$

Step 5.5

Subtract $y$ from $4 y$ .

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

Step 5.6

Factor.

Tap for more steps...

Step 5.6.1

Factor $y^{2} + 3 y - 4$ using the AC method.

Tap for more steps...

Step 5.6.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 5.6.1.2

Write the factored form using these integers.

$(y + 4) (y + 1) ((y - 1) (y + 4))$

Step 5.6.2

Remove unnecessary parentheses.

$(y + 4) (y + 1) (y - 1) (y + 4)$

Step 5.7

Combine exponents.

Tap for more steps...

Step 5.7.1

Raise $y + 4$ to the power of $1$ .

${(y + 4)}^{1} (y + 4) (y + 1) (y - 1)$

Step 5.7.2

Raise $y + 4$ to the power of $1$ .

${(y + 4)}^{1} {(y + 4)}^{1} (y + 1) (y - 1)$

Step 5.7.3

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

${(y + 4)}^{1 + 1} (y + 1) (y - 1)$

Step 5.7.4

Add $1$ and $1$ .

${(y + 4)}^{2} (y + 1) (y - 1)$