Finite Math Examples

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

Step 1

Rewrite ${(- \frac{2 y}{3} + \frac{1}{3})}^{2}$ as $(- \frac{2 y}{3} + \frac{1}{3}) (- \frac{2 y}{3} + \frac{1}{3})$ .

$3 ((- \frac{2 y}{3} + \frac{1}{3}) (- \frac{2 y}{3} + \frac{1}{3}))$

Step 2

Expand $(- \frac{2 y}{3} + \frac{1}{3}) (- \frac{2 y}{3} + \frac{1}{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$3 (- \frac{2 y}{3} (- \frac{2 y}{3} + \frac{1}{3}) + \frac{1}{3} (- \frac{2 y}{3} + \frac{1}{3}))$

Step 2.2

Apply the distributive property.

$3 (- \frac{2 y}{3} (- \frac{2 y}{3}) - \frac{2 y}{3} \cdot \frac{1}{3} + \frac{1}{3} (- \frac{2 y}{3} + \frac{1}{3}))$

Step 2.3

Apply the distributive property.

$3 (- \frac{2 y}{3} (- \frac{2 y}{3}) - \frac{2 y}{3} \cdot \frac{1}{3} + \frac{1}{3} (- \frac{2 y}{3}) + \frac{1}{3} \cdot \frac{1}{3})$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $- \frac{2 y}{3} (- \frac{2 y}{3})$ .

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Step 3.1.1.1

Multiply $- 1$ by $- 1$ .

$3 (1 \frac{2 y}{3} \frac{2 y}{3} - \frac{2 y}{3} \cdot \frac{1}{3} + \frac{1}{3} (- \frac{2 y}{3}) + \frac{1}{3} \cdot \frac{1}{3})$

Step 3.1.1.2

Multiply $\frac{2 y}{3}$ by $1$ .

$3 (\frac{2 y}{3} \cdot \frac{2 y}{3} - \frac{2 y}{3} \cdot \frac{1}{3} + \frac{1}{3} (- \frac{2 y}{3}) + \frac{1}{3} \cdot \frac{1}{3})$

Step 3.1.1.3

Multiply $\frac{2 y}{3}$ by $\frac{2 y}{3}$ .

$3 (\frac{2 y (2 y)}{3 \cdot 3} - \frac{2 y}{3} \cdot \frac{1}{3} + \frac{1}{3} (- \frac{2 y}{3}) + \frac{1}{3} \cdot \frac{1}{3})$

Step 3.1.1.4

Multiply $2$ by $2$ .

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

Step 3.1.1.5

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

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

Step 3.1.1.6

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

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

Step 3.1.1.7

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

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

Step 3.1.1.8

Add $1$ and $1$ .

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

Step 3.1.1.9

Multiply $3$ by $3$ .

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

Step 3.1.2

Multiply $- \frac{2 y}{3} \cdot \frac{1}{3}$ .

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Step 3.1.2.1

Multiply $\frac{1}{3}$ by $\frac{2 y}{3}$ .

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

Step 3.1.2.2

Multiply $3$ by $3$ .

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

Step 3.1.3

Multiply $\frac{1}{3} (- \frac{2 y}{3})$ .

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Step 3.1.3.1

Multiply $\frac{1}{3}$ by $\frac{2 y}{3}$ .

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

Step 3.1.3.2

Multiply $3$ by $3$ .

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

Step 3.1.4

Multiply $\frac{1}{3} \cdot \frac{1}{3}$ .

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Step 3.1.4.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

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

Step 3.1.4.2

Multiply $3$ by $3$ .

$3 (\frac{4 y^{2}}{9} - \frac{2 y}{9} - \frac{2 y}{9} + \frac{1}{9})$

Step 3.2

Subtract $\frac{2 y}{9}$ from $- \frac{2 y}{9}$ .

$3 (\frac{4 y^{2}}{9} - 2 \frac{2 y}{9} + \frac{1}{9})$

Step 4

Simplify each term.

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Step 4.1

Multiply $- 2 \frac{2 y}{9}$ .

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Step 4.1.1

Combine $- 2$ and $\frac{2 y}{9}$ .

$3 (\frac{4 y^{2}}{9} + \frac{- 2 (2 y)}{9} + \frac{1}{9})$

Step 4.1.2

Multiply $2$ by $- 2$ .

$3 (\frac{4 y^{2}}{9} + \frac{- 4 y}{9} + \frac{1}{9})$

Step 4.2

Move the negative in front of the fraction.

$3 (\frac{4 y^{2}}{9} - \frac{4 y}{9} + \frac{1}{9})$

Step 5

Apply the distributive property.

$3 \frac{4 y^{2}}{9} + 3 (- \frac{4 y}{9}) + 3 (\frac{1}{9})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $3$ .

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Step 6.1.1

Factor $3$ out of $9$ .

$3 \frac{4 y^{2}}{3 (3)} + 3 (- \frac{4 y}{9}) + 3 (\frac{1}{9})$

Step 6.1.2

Cancel the common factor.

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

Step 6.1.3

Rewrite the expression.

$\frac{4 y^{2}}{3} + 3 (- \frac{4 y}{9}) + 3 (\frac{1}{9})$

Step 6.2

Cancel the common factor of $3$ .

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Step 6.2.1

Move the leading negative in $- \frac{4 y}{9}$ into the numerator.

$\frac{4 y^{2}}{3} + 3 \frac{- 4 y}{9} + 3 (\frac{1}{9})$

Step 6.2.2

Factor $3$ out of $9$ .

$\frac{4 y^{2}}{3} + 3 \frac{- 4 y}{3 (3)} + 3 (\frac{1}{9})$

Step 6.2.3

Cancel the common factor.

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

Step 6.2.4

Rewrite the expression.

$\frac{4 y^{2}}{3} + \frac{- 4 y}{3} + 3 (\frac{1}{9})$

Step 6.3

Cancel the common factor of $3$ .

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Step 6.3.1

Factor $3$ out of $9$ .

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

Step 6.3.2

Cancel the common factor.

$\frac{4 y^{2}}{3} + \frac{- 4 y}{3} + 3 \frac{1}{3 \cdot 3}$

Step 6.3.3

Rewrite the expression.

$\frac{4 y^{2}}{3} + \frac{- 4 y}{3} + \frac{1}{3}$

Step 7

Move the negative in front of the fraction.

$\frac{4 y^{2}}{3} - \frac{4 y}{3} + \frac{1}{3}$

Step 8

Factor out the GCF of $\frac{1}{3}$ from $\frac{4 y^{2}}{3} - \frac{4 y}{3} + \frac{1}{3}$ .

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Step 8.1

Factor out the GCF of $\frac{1}{3}$ from each term in the polynomial.

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Step 8.1.1

Factor out the GCF of $\frac{1}{3}$ from the expression $\frac{4 y^{2}}{3}$ .

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

Step 8.1.2

Factor out the GCF of $\frac{1}{3}$ from the expression $- \frac{4 y}{3}$ .

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

Step 8.1.3

Factor out the GCF of $\frac{1}{3}$ from the expression $\frac{1}{3}$ .

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

Step 8.2

Since all the terms share a common factor of $\frac{1}{3}$ , it can be factored out of each term.

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

Step 9

The polynomial cannot be factored using the specified method. Try a different method, or if you aren't sure, choose Factor.

The polynomial cannot be factored using the specified method.