Algebra Examples

$- 2 y (y^{2} - 6 y - 3) = a y^{3} + 12 y^{2} + b y$

Step 1

Rewrite the equation as $a y^{3} + 12 y^{2} + b y = - 2 y (y^{2} - 6 y - 3)$ .

$a y^{3} + 12 y^{2} + b y = - 2 y (y^{2} - 6 y - 3)$

Step 2

Simplify $- 2 y (y^{2} - 6 y - 3)$ .

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Step 2.1

Rewrite.

$a y^{3} + 12 y^{2} + b y = 0 + 0 - 2 y (y^{2} - 6 y - 3)$

Step 2.2

Simplify by adding zeros.

$a y^{3} + 12 y^{2} + b y = - 2 y (y^{2} - 6 y - 3)$

Step 2.3

Apply the distributive property.

$a y^{3} + 12 y^{2} + b y = - 2 y \cdot y^{2} - 2 y (- 6 y) - 2 y \cdot - 3$

Step 2.4

Simplify.

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Step 2.4.1

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

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Step 2.4.1.1

Move $y^{2}$ .

$a y^{3} + 12 y^{2} + b y = - 2 (y^{2} y) - 2 y (- 6 y) - 2 y \cdot - 3$

Step 2.4.1.2

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

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Step 2.4.1.2.1

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

$a y^{3} + 12 y^{2} + b y = - 2 (y^{2} y^{1}) - 2 y (- 6 y) - 2 y \cdot - 3$

Step 2.4.1.2.2

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

$a y^{3} + 12 y^{2} + b y = - 2 y^{2 + 1} - 2 y (- 6 y) - 2 y \cdot - 3$

Step 2.4.1.3

Add $2$ and $1$ .

$a y^{3} + 12 y^{2} + b y = - 2 y^{3} - 2 y (- 6 y) - 2 y \cdot - 3$

Step 2.4.2

Rewrite using the commutative property of multiplication.

$a y^{3} + 12 y^{2} + b y = - 2 y^{3} - 2 \cdot - 6 y \cdot y - 2 y \cdot - 3$

Step 2.4.3

Multiply $- 3$ by $- 2$ .

$a y^{3} + 12 y^{2} + b y = - 2 y^{3} - 2 \cdot - 6 y \cdot y + 6 y$

Step 2.5

Simplify each term.

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Step 2.5.1

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

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Step 2.5.1.1

Move $y$ .

$a y^{3} + 12 y^{2} + b y = - 2 y^{3} - 2 \cdot - 6 (y \cdot y) + 6 y$

Step 2.5.1.2

Multiply $y$ by $y$ .

$a y^{3} + 12 y^{2} + b y = - 2 y^{3} - 2 \cdot - 6 y^{2} + 6 y$

Step 2.5.2

Multiply $- 2$ by $- 6$ .

$a y^{3} + 12 y^{2} + b y = - 2 y^{3} + 12 y^{2} + 6 y$

Step 3

Move all terms not containing $a$ to the right side of the equation.

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Step 3.1

Subtract $12 y^{2}$ from both sides of the equation.

$a y^{3} + b y = - 2 y^{3} + 12 y^{2} + 6 y - 12 y^{2}$

Step 3.2

Subtract $b y$ from both sides of the equation.

$a y^{3} = - 2 y^{3} + 12 y^{2} + 6 y - 12 y^{2} - b y$

Step 3.3

Combine the opposite terms in $- 2 y^{3} + 12 y^{2} + 6 y - 12 y^{2} - b y$ .

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Step 3.3.1

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

$a y^{3} = - 2 y^{3} + 6 y + 0 - b y$

Step 3.3.2

Add $- 2 y^{3} + 6 y$ and $0$ .

$a y^{3} = - 2 y^{3} + 6 y - b y$

Step 4

Divide each term in $a y^{3} = - 2 y^{3} + 6 y - b y$ by $y^{3}$ and simplify.

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Step 4.1

Divide each term in $a y^{3} = - 2 y^{3} + 6 y - b y$ by $y^{3}$ .

$\frac{a y^{3}}{y^{3}} = \frac{- 2 y^{3}}{y^{3}} + \frac{6 y}{y^{3}} + \frac{- b y}{y^{3}}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $y^{3}$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{a y^{3}}{y^{3}} = \frac{- 2 y^{3}}{y^{3}} + \frac{6 y}{y^{3}} + \frac{- b y}{y^{3}}$

Step 4.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 2 y^{3}}{y^{3}} + \frac{6 y}{y^{3}} + \frac{- b y}{y^{3}}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Cancel the common factor of $y^{3}$ .

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Step 4.3.1.1.1

Cancel the common factor.

$a = \frac{- 2 y^{3}}{y^{3}} + \frac{6 y}{y^{3}} + \frac{- b y}{y^{3}}$

Step 4.3.1.1.2

Divide $- 2$ by $1$ .

$a = - 2 + \frac{6 y}{y^{3}} + \frac{- b y}{y^{3}}$

Step 4.3.1.2

Cancel the common factor of $y$ and $y^{3}$ .

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Step 4.3.1.2.1

Factor $y$ out of $6 y$ .

$a = - 2 + \frac{y \cdot 6}{y^{3}} + \frac{- b y}{y^{3}}$

Step 4.3.1.2.2

Cancel the common factors.

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Step 4.3.1.2.2.1

Factor $y$ out of $y^{3}$ .

$a = - 2 + \frac{y \cdot 6}{y \cdot y^{2}} + \frac{- b y}{y^{3}}$

Step 4.3.1.2.2.2

Cancel the common factor.

$a = - 2 + \frac{y \cdot 6}{y \cdot y^{2}} + \frac{- b y}{y^{3}}$

Step 4.3.1.2.2.3

Rewrite the expression.

$a = - 2 + \frac{6}{y^{2}} + \frac{- b y}{y^{3}}$

Step 4.3.1.3

Cancel the common factor of $y$ and $y^{3}$ .

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Step 4.3.1.3.1

Factor $y$ out of $- b y$ .

$a = - 2 + \frac{6}{y^{2}} + \frac{y (- b)}{y^{3}}$

Step 4.3.1.3.2

Cancel the common factors.

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Step 4.3.1.3.2.1

Factor $y$ out of $y^{3}$ .

$a = - 2 + \frac{6}{y^{2}} + \frac{y (- b)}{y \cdot y^{2}}$

Step 4.3.1.3.2.2

Cancel the common factor.

$a = - 2 + \frac{6}{y^{2}} + \frac{y (- b)}{y \cdot y^{2}}$

Step 4.3.1.3.2.3

Rewrite the expression.

$a = - 2 + \frac{6}{y^{2}} + \frac{- b}{y^{2}}$

Step 4.3.1.4

Move the negative in front of the fraction.

$a = - 2 + \frac{6}{y^{2}} - \frac{b}{y^{2}}$