Algebra Examples

$(y - 3) (y^{2} + 9) (y + 3) - {(2 y^{2} - y)}^{2} - 19$

Step 1

Simplify each term.

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Step 1.1

Expand $(y - 3) (y^{2} + 9)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$(y (y^{2} + 9) - 3 (y^{2} + 9)) (y + 3) - {(2 y^{2} - y)}^{2} - 19$

Step 1.1.2

Apply the distributive property.

$(y \cdot y^{2} + y \cdot 9 - 3 (y^{2} + 9)) (y + 3) - {(2 y^{2} - y)}^{2} - 19$

Step 1.1.3

Apply the distributive property.

$(y \cdot y^{2} + y \cdot 9 - 3 y^{2} - 3 \cdot 9) (y + 3) - {(2 y^{2} - y)}^{2} - 19$

Step 1.2

Simplify each term.

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Step 1.2.1

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

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Step 1.2.1.1

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

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Step 1.2.1.1.1

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

$(y^{1} y^{2} + y \cdot 9 - 3 y^{2} - 3 \cdot 9) (y + 3) - {(2 y^{2} - y)}^{2} - 19$

Step 1.2.1.1.2

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

$(y^{1 + 2} + y \cdot 9 - 3 y^{2} - 3 \cdot 9) (y + 3) - {(2 y^{2} - y)}^{2} - 19$

Step 1.2.1.2

Add $1$ and $2$ .

$(y^{3} + y \cdot 9 - 3 y^{2} - 3 \cdot 9) (y + 3) - {(2 y^{2} - y)}^{2} - 19$

Step 1.2.2

Move $9$ to the left of $y$ .

$(y^{3} + 9 \cdot y - 3 y^{2} - 3 \cdot 9) (y + 3) - {(2 y^{2} - y)}^{2} - 19$

Step 1.2.3

Multiply $- 3$ by $9$ .

$(y^{3} + 9 y - 3 y^{2} - 27) (y + 3) - {(2 y^{2} - y)}^{2} - 19$

Step 1.3

Expand $(y^{3} + 9 y - 3 y^{2} - 27) (y + 3)$ by multiplying each term in the first expression by each term in the second expression.

$y^{3} y + y^{3} \cdot 3 + 9 y \cdot y + 9 y \cdot 3 - 3 y^{2} y - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $y^{3}$ by $y$ by adding the exponents.

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Step 1.4.1.1

Multiply $y^{3}$ by $y$ .

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Step 1.4.1.1.1

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

$y^{3} y^{1} + y^{3} \cdot 3 + 9 y \cdot y + 9 y \cdot 3 - 3 y^{2} y - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.1.1.2

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

$y^{3 + 1} + y^{3} \cdot 3 + 9 y \cdot y + 9 y \cdot 3 - 3 y^{2} y - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.1.2

Add $3$ and $1$ .

$y^{4} + y^{3} \cdot 3 + 9 y \cdot y + 9 y \cdot 3 - 3 y^{2} y - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.2

Move $3$ to the left of $y^{3}$ .

$y^{4} + 3 \cdot y^{3} + 9 y \cdot y + 9 y \cdot 3 - 3 y^{2} y - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.3

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

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Step 1.4.3.1

Move $y$ .

$y^{4} + 3 y^{3} + 9 (y \cdot y) + 9 y \cdot 3 - 3 y^{2} y - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.3.2

Multiply $y$ by $y$ .

$y^{4} + 3 y^{3} + 9 y^{2} + 9 y \cdot 3 - 3 y^{2} y - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.4

Multiply $3$ by $9$ .

$y^{4} + 3 y^{3} + 9 y^{2} + 27 y - 3 y^{2} y - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.5

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

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Step 1.4.5.1

Move $y$ .

$y^{4} + 3 y^{3} + 9 y^{2} + 27 y - 3 (y \cdot y^{2}) - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.5.2

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

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Step 1.4.5.2.1

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

$y^{4} + 3 y^{3} + 9 y^{2} + 27 y - 3 (y^{1} y^{2}) - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.5.2.2

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

$y^{4} + 3 y^{3} + 9 y^{2} + 27 y - 3 y^{1 + 2} - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.5.3

Add $1$ and $2$ .

$y^{4} + 3 y^{3} + 9 y^{2} + 27 y - 3 y^{3} - 3 y^{2} \cdot 3 - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.6

Multiply $3$ by $- 3$ .

$y^{4} + 3 y^{3} + 9 y^{2} + 27 y - 3 y^{3} - 9 y^{2} - 27 y - 27 \cdot 3 - {(2 y^{2} - y)}^{2} - 19$

Step 1.4.7

Multiply $- 27$ by $3$ .

$y^{4} + 3 y^{3} + 9 y^{2} + 27 y - 3 y^{3} - 9 y^{2} - 27 y - 81 - {(2 y^{2} - y)}^{2} - 19$

Step 1.5

Combine the opposite terms in $y^{4} + 3 y^{3} + 9 y^{2} + 27 y - 3 y^{3} - 9 y^{2} - 27 y - 81$ .

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Step 1.5.1

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

$y^{4} + 9 y^{2} + 27 y + 0 - 9 y^{2} - 27 y - 81 - {(2 y^{2} - y)}^{2} - 19$

Step 1.5.2

Add $y^{4} + 9 y^{2} + 27 y$ and $0$ .

$y^{4} + 9 y^{2} + 27 y - 9 y^{2} - 27 y - 81 - {(2 y^{2} - y)}^{2} - 19$

Step 1.5.3

Subtract $9 y^{2}$ from $9 y^{2}$ .

$y^{4} + 27 y + 0 - 27 y - 81 - {(2 y^{2} - y)}^{2} - 19$

Step 1.5.4

Add $y^{4} + 27 y$ and $0$ .

$y^{4} + 27 y - 27 y - 81 - {(2 y^{2} - y)}^{2} - 19$

Step 1.5.5

Subtract $27 y$ from $27 y$ .

$y^{4} + 0 - 81 - {(2 y^{2} - y)}^{2} - 19$

Step 1.5.6

Add $y^{4}$ and $0$ .

$y^{4} - 81 - {(2 y^{2} - y)}^{2} - 19$

Step 1.6

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

$y^{4} - 81 - ((2 y^{2} - y) (2 y^{2} - y)) - 19$

Step 1.7

Expand $(2 y^{2} - y) (2 y^{2} - y)$ using the FOIL Method.

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Step 1.7.1

Apply the distributive property.

$y^{4} - 81 - (2 y^{2} (2 y^{2} - y) - y (2 y^{2} - y)) - 19$

Step 1.7.2

Apply the distributive property.

$y^{4} - 81 - (2 y^{2} (2 y^{2}) + 2 y^{2} (- y) - y (2 y^{2} - y)) - 19$

Step 1.7.3

Apply the distributive property.

$y^{4} - 81 - (2 y^{2} (2 y^{2}) + 2 y^{2} (- y) - y (2 y^{2}) - y (- y)) - 19$

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Rewrite using the commutative property of multiplication.

$y^{4} - 81 - (2 \cdot 2 y^{2} y^{2} + 2 y^{2} (- y) - y (2 y^{2}) - y (- y)) - 19$

Step 1.8.1.2

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

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Step 1.8.1.2.1

Move $y^{2}$ .

$y^{4} - 81 - (2 \cdot 2 (y^{2} y^{2}) + 2 y^{2} (- y) - y (2 y^{2}) - y (- y)) - 19$

Step 1.8.1.2.2

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

$y^{4} - 81 - (2 \cdot 2 y^{2 + 2} + 2 y^{2} (- y) - y (2 y^{2}) - y (- y)) - 19$

Step 1.8.1.2.3

Add $2$ and $2$ .

$y^{4} - 81 - (2 \cdot 2 y^{4} + 2 y^{2} (- y) - y (2 y^{2}) - y (- y)) - 19$

Step 1.8.1.3

Multiply $2$ by $2$ .

$y^{4} - 81 - (4 y^{4} + 2 y^{2} (- y) - y (2 y^{2}) - y (- y)) - 19$

Step 1.8.1.4

Rewrite using the commutative property of multiplication.

$y^{4} - 81 - (4 y^{4} + 2 \cdot - 1 y^{2} y - y (2 y^{2}) - y (- y)) - 19$

Step 1.8.1.5

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

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Step 1.8.1.5.1

Move $y$ .

$y^{4} - 81 - (4 y^{4} + 2 \cdot - 1 (y \cdot y^{2}) - y (2 y^{2}) - y (- y)) - 19$

Step 1.8.1.5.2

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

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Step 1.8.1.5.2.1

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

$y^{4} - 81 - (4 y^{4} + 2 \cdot - 1 (y^{1} y^{2}) - y (2 y^{2}) - y (- y)) - 19$

Step 1.8.1.5.2.2

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

$y^{4} - 81 - (4 y^{4} + 2 \cdot - 1 y^{1 + 2} - y (2 y^{2}) - y (- y)) - 19$

Step 1.8.1.5.3

Add $1$ and $2$ .

$y^{4} - 81 - (4 y^{4} + 2 \cdot - 1 y^{3} - y (2 y^{2}) - y (- y)) - 19$

Step 1.8.1.6

Multiply $2$ by $- 1$ .

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - y (2 y^{2}) - y (- y)) - 19$

Step 1.8.1.7

Rewrite using the commutative property of multiplication.

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - 1 \cdot 2 y \cdot y^{2} - y (- y)) - 19$

Step 1.8.1.8

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

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Step 1.8.1.8.1

Move $y^{2}$ .

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - 1 \cdot 2 (y^{2} y) - y (- y)) - 19$

Step 1.8.1.8.2

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

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Step 1.8.1.8.2.1

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

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - 1 \cdot 2 (y^{2} y^{1}) - y (- y)) - 19$

Step 1.8.1.8.2.2

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

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - 1 \cdot 2 y^{2 + 1} - y (- y)) - 19$

Step 1.8.1.8.3

Add $2$ and $1$ .

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - 1 \cdot 2 y^{3} - y (- y)) - 19$

Step 1.8.1.9

Multiply $- 1$ by $2$ .

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - 2 y^{3} - y (- y)) - 19$

Step 1.8.1.10

Rewrite using the commutative property of multiplication.

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - 2 y^{3} - 1 \cdot - 1 y \cdot y) - 19$

Step 1.8.1.11

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

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Step 1.8.1.11.1

Move $y$ .

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - 2 y^{3} - 1 \cdot - 1 (y \cdot y)) - 19$

Step 1.8.1.11.2

Multiply $y$ by $y$ .

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - 2 y^{3} - 1 \cdot - 1 y^{2}) - 19$

Step 1.8.1.12

Multiply $- 1$ by $- 1$ .

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - 2 y^{3} + 1 y^{2}) - 19$

Step 1.8.1.13

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

$y^{4} - 81 - (4 y^{4} - 2 y^{3} - 2 y^{3} + y^{2}) - 19$

Step 1.8.2

Subtract $2 y^{3}$ from $- 2 y^{3}$ .

$y^{4} - 81 - (4 y^{4} - 4 y^{3} + y^{2}) - 19$

Step 1.9

Apply the distributive property.

$y^{4} - 81 - (4 y^{4}) - (- 4 y^{3}) - y^{2} - 19$

Step 1.10

Simplify.

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Step 1.10.1

Multiply $4$ by $- 1$ .

$y^{4} - 81 - 4 y^{4} - (- 4 y^{3}) - y^{2} - 19$

Step 1.10.2

Multiply $- 4$ by $- 1$ .

$y^{4} - 81 - 4 y^{4} + 4 y^{3} - y^{2} - 19$

Step 2

Simplify by adding terms.

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Step 2.1

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

$- 3 y^{4} - 81 + 4 y^{3} - y^{2} - 19$

Step 2.2

Subtract $19$ from $- 81$ .

$- 3 y^{4} + 4 y^{3} - y^{2} - 100$