Finite Math Examples

$(y + 2) (y - 6) (y^{2} - 4 y + 12)$

Step 1

Expand $(y + 2) (y - 6)$ using the FOIL Method.

Tap for more steps...

Step 1.1

Apply the distributive property.

$(y (y - 6) + 2 (y - 6)) (y^{2} - 4 y + 12)$

Step 1.2

Apply the distributive property.

$(y \cdot y + y \cdot - 6 + 2 (y - 6)) (y^{2} - 4 y + 12)$

Step 1.3

Apply the distributive property.

$(y \cdot y + y \cdot - 6 + 2 y + 2 \cdot - 6) (y^{2} - 4 y + 12)$

Step 2

Simplify and combine like terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Multiply $y$ by $y$ .

$(y^{2} + y \cdot - 6 + 2 y + 2 \cdot - 6) (y^{2} - 4 y + 12)$

Step 2.1.2

Move $- 6$ to the left of $y$ .

$(y^{2} - 6 \cdot y + 2 y + 2 \cdot - 6) (y^{2} - 4 y + 12)$

Step 2.1.3

Multiply $2$ by $- 6$ .

$(y^{2} - 6 y + 2 y - 12) (y^{2} - 4 y + 12)$

Step 2.2

Add $- 6 y$ and $2 y$ .

$(y^{2} - 4 y - 12) (y^{2} - 4 y + 12)$

Step 3

Expand $(y^{2} - 4 y - 12) (y^{2} - 4 y + 12)$ by multiplying each term in the first expression by each term in the second expression.

$y^{2} y^{2} + y^{2} (- 4 y) + y^{2} \cdot 12 - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 - 12 y^{2} - 12 (- 4 y) - 12 \cdot 12$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Combine the opposite terms in $y^{2} y^{2} + y^{2} (- 4 y) + y^{2} \cdot 12 - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 - 12 y^{2} - 12 (- 4 y) - 12 \cdot 12$ .

Tap for more steps...

Step 4.1.1

Reorder the factors in the terms $y^{2} \cdot 12$ and $- 12 y^{2}$ .

$y^{2} y^{2} + y^{2} (- 4 y) + 12 y^{2} - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 - 12 y^{2} - 12 (- 4 y) - 12 \cdot 12$

Step 4.1.2

Subtract $12 y^{2}$ from $12 y^{2}$ .

$y^{2} y^{2} + y^{2} (- 4 y) - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 + 0 - 12 (- 4 y) - 12 \cdot 12$

Step 4.1.3

Add $y^{2} y^{2} + y^{2} (- 4 y) - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12$ and $0$ .

$y^{2} y^{2} + y^{2} (- 4 y) - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2

Simplify each term.

Tap for more steps...

Step 4.2.1

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

Tap for more steps...

Step 4.2.1.1

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

$y^{2 + 2} + y^{2} (- 4 y) - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.1.2

Add $2$ and $2$ .

$y^{4} + y^{2} (- 4 y) - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.2

Rewrite using the commutative property of multiplication.

$y^{4} - 4 y^{2} y - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.3

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

Tap for more steps...

Step 4.2.3.1

Move $y$ .

$y^{4} - 4 (y \cdot y^{2}) - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.3.2

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

Tap for more steps...

Step 4.2.3.2.1

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

$y^{4} - 4 (y^{1} y^{2}) - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.3.2.2

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

$y^{4} - 4 y^{1 + 2} - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.3.3

Add $1$ and $2$ .

$y^{4} - 4 y^{3} - 4 y \cdot y^{2} - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.4

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

Tap for more steps...

Step 4.2.4.1

Move $y^{2}$ .

$y^{4} - 4 y^{3} - 4 (y^{2} y) - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.4.2

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

Tap for more steps...

Step 4.2.4.2.1

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

$y^{4} - 4 y^{3} - 4 (y^{2} y^{1}) - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.4.2.2

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

$y^{4} - 4 y^{3} - 4 y^{2 + 1} - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.4.3

Add $2$ and $1$ .

$y^{4} - 4 y^{3} - 4 y^{3} - 4 y (- 4 y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.5

Rewrite using the commutative property of multiplication.

$y^{4} - 4 y^{3} - 4 y^{3} - 4 \cdot - 4 y \cdot y - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.6

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 4.2.6.1

Move $y$ .

$y^{4} - 4 y^{3} - 4 y^{3} - 4 \cdot - 4 (y \cdot y) - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.6.2

Multiply $y$ by $y$ .

$y^{4} - 4 y^{3} - 4 y^{3} - 4 \cdot - 4 y^{2} - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.7

Multiply $- 4$ by $- 4$ .

$y^{4} - 4 y^{3} - 4 y^{3} + 16 y^{2} - 4 y \cdot 12 - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.8

Multiply $12$ by $- 4$ .

$y^{4} - 4 y^{3} - 4 y^{3} + 16 y^{2} - 48 y - 12 (- 4 y) - 12 \cdot 12$

Step 4.2.9

Multiply $- 4$ by $- 12$ .

$y^{4} - 4 y^{3} - 4 y^{3} + 16 y^{2} - 48 y + 48 y - 12 \cdot 12$

Step 4.2.10

Multiply $- 12$ by $12$ .

$y^{4} - 4 y^{3} - 4 y^{3} + 16 y^{2} - 48 y + 48 y - 144$

Step 4.3

Simplify by adding terms.

Tap for more steps...

Step 4.3.1

Combine the opposite terms in $y^{4} - 4 y^{3} - 4 y^{3} + 16 y^{2} - 48 y + 48 y - 144$ .

Tap for more steps...

Step 4.3.1.1

Add $- 48 y$ and $48 y$ .

$y^{4} - 4 y^{3} - 4 y^{3} + 16 y^{2} + 0 - 144$

Step 4.3.1.2

Add $y^{4} - 4 y^{3} - 4 y^{3} + 16 y^{2}$ and $0$ .

$y^{4} - 4 y^{3} - 4 y^{3} + 16 y^{2} - 144$

Step 4.3.2

Subtract $4 y^{3}$ from $- 4 y^{3}$ .

$y^{4} - 8 y^{3} + 16 y^{2} - 144$

Step 5

Factor using the perfect square rule.

Tap for more steps...

Step 5.1

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

${(y^{2})}^{2} - 8 y^{3} + 16 y^{2} - 144$

Step 5.2

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

${(y^{2})}^{2} - 8 y^{3} + {(4 y)}^{2} - 144$

Step 5.3

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 5.4

Rewrite the polynomial.

${(y^{2})}^{2} - 2 \cdot y^{2} \cdot (4 y) + {(4 y)}^{2} - 144$

Step 5.5

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

${(y^{2} - 4 y)}^{2} - 144$

Step 6

Rewrite $144$ as $12^{2}$ .

${(y^{2} - 4 y)}^{2} - 12^{2}$

Step 7

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 $b = 12$ .

$(y^{2} - 4 y + 12) (y^{2} - 4 y - 12)$

Step 8

Factor.

Tap for more steps...

Step 8.1

Factor $y^{2} - 4 y - 12$ using the AC method.

Tap for more steps...

Step 8.1.1

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 $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Step 8.1.2

Write the factored form using these integers.

$(y^{2} - 4 y + 12) ((y - 6) (y + 2))$

Step 8.2

Remove unnecessary parentheses.

$(y^{2} - 4 y + 12) (y - 6) (y + 2)$