Basic Math Examples

$(3 v - \frac{1}{2} w) (v - \frac{1}{5} w)$

Step 1

Combine fractions.

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Step 1.1

Combine $w$ and $\frac{1}{2}$ .

$(3 v - \frac{w}{2}) (v - \frac{1}{5} w)$

Step 1.2

Combine $w$ and $\frac{1}{5}$ .

$(3 v - \frac{w}{2}) (v - \frac{w}{5})$

Step 2

Expand $(3 v - \frac{w}{2}) (v - \frac{w}{5})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$3 v (v - \frac{w}{5}) - \frac{w}{2} (v - \frac{w}{5})$

Step 2.2

Apply the distributive property.

$3 v \cdot v + 3 v (- \frac{w}{5}) - \frac{w}{2} (v - \frac{w}{5})$

Step 2.3

Apply the distributive property.

$3 v \cdot v + 3 v (- \frac{w}{5}) - \frac{w}{2} v - \frac{w}{2} (- \frac{w}{5})$

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 $v$ by $v$ by adding the exponents.

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Step 3.1.1.1

Move $v$ .

$3 (v \cdot v) + 3 v (- \frac{w}{5}) - \frac{w}{2} v - \frac{w}{2} (- \frac{w}{5})$

Step 3.1.1.2

Multiply $v$ by $v$ .

$3 v^{2} + 3 v (- \frac{w}{5}) - \frac{w}{2} v - \frac{w}{2} (- \frac{w}{5})$

Step 3.1.2

Multiply $3 v (- \frac{w}{5})$ .

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Step 3.1.2.1

Multiply $- 1$ by $3$ .

$3 v^{2} - 3 v \frac{w}{5} - \frac{w}{2} v - \frac{w}{2} (- \frac{w}{5})$

Step 3.1.2.2

Combine $- 3$ and $\frac{w}{5}$ .

$3 v^{2} + v \frac{- 3 w}{5} - \frac{w}{2} v - \frac{w}{2} (- \frac{w}{5})$

Step 3.1.2.3

Combine $v$ and $\frac{- 3 w}{5}$ .

$3 v^{2} + \frac{v (- 3 w)}{5} - \frac{w}{2} v - \frac{w}{2} (- \frac{w}{5})$

Step 3.1.3

Remove unnecessary parentheses.

$3 v^{2} + \frac{v \cdot - 3 w}{5} - \frac{w}{2} v - \frac{w}{2} (- \frac{w}{5})$

Step 3.1.4

Move $- 3$ to the left of $v$ .

$3 v^{2} + \frac{- 3 \cdot v w}{5} - \frac{w}{2} v - \frac{w}{2} (- \frac{w}{5})$

Step 3.1.5

Move the negative in front of the fraction.

$3 v^{2} - \frac{3 \cdot v w}{5} - \frac{w}{2} v - \frac{w}{2} (- \frac{w}{5})$

Step 3.1.6

Combine $v$ and $\frac{w}{2}$ .

$3 v^{2} - \frac{3 v w}{5} - \frac{v w}{2} - \frac{w}{2} (- \frac{w}{5})$

Step 3.1.7

Multiply $- \frac{w}{2} (- \frac{w}{5})$ .

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Step 3.1.7.1

Multiply $- 1$ by $- 1$ .

$3 v^{2} - \frac{3 v w}{5} - \frac{v w}{2} + 1 \frac{w}{2} \frac{w}{5}$

Step 3.1.7.2

Multiply $\frac{w}{2}$ by $1$ .

$3 v^{2} - \frac{3 v w}{5} - \frac{v w}{2} + \frac{w}{2} \cdot \frac{w}{5}$

Step 3.1.7.3

Multiply $\frac{w}{2}$ by $\frac{w}{5}$ .

$3 v^{2} - \frac{3 v w}{5} - \frac{v w}{2} + \frac{w \cdot w}{2 \cdot 5}$

Step 3.1.7.4

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

$3 v^{2} - \frac{3 v w}{5} - \frac{v w}{2} + \frac{w^{1} w}{2 \cdot 5}$

Step 3.1.7.5

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

$3 v^{2} - \frac{3 v w}{5} - \frac{v w}{2} + \frac{w^{1} w^{1}}{2 \cdot 5}$

Step 3.1.7.6

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

$3 v^{2} - \frac{3 v w}{5} - \frac{v w}{2} + \frac{w^{1 + 1}}{2 \cdot 5}$

Step 3.1.7.7

Add $1$ and $1$ .

$3 v^{2} - \frac{3 v w}{5} - \frac{v w}{2} + \frac{w^{2}}{2 \cdot 5}$

Step 3.1.7.8

Multiply $2$ by $5$ .

$3 v^{2} - \frac{3 v w}{5} - \frac{v w}{2} + \frac{w^{2}}{10}$

Step 3.2

To write $- \frac{3 v w}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 v^{2} - \frac{3 v w}{5} \cdot \frac{2}{2} - \frac{v w}{2} + \frac{w^{2}}{10}$

Step 3.3

To write $- \frac{v w}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$3 v^{2} - \frac{3 v w}{5} \cdot \frac{2}{2} - \frac{v w}{2} \cdot \frac{5}{5} + \frac{w^{2}}{10}$

Step 3.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{3 v w}{5}$ by $\frac{2}{2}$ .

$3 v^{2} - \frac{3 v w \cdot 2}{5 \cdot 2} - \frac{v w}{2} \cdot \frac{5}{5} + \frac{w^{2}}{10}$

Step 3.4.2

Multiply $5$ by $2$ .

$3 v^{2} - \frac{3 v w \cdot 2}{10} - \frac{v w}{2} \cdot \frac{5}{5} + \frac{w^{2}}{10}$

Step 3.4.3

Multiply $\frac{v w}{2}$ by $\frac{5}{5}$ .

$3 v^{2} - \frac{3 v w \cdot 2}{10} - \frac{v w \cdot 5}{2 \cdot 5} + \frac{w^{2}}{10}$

Step 3.4.4

Multiply $2$ by $5$ .

$3 v^{2} - \frac{3 v w \cdot 2}{10} - \frac{v w \cdot 5}{10} + \frac{w^{2}}{10}$

Step 3.5

Combine the numerators over the common denominator.

$3 v^{2} + \frac{- 3 v w \cdot 2 - v w \cdot 5}{10} + \frac{w^{2}}{10}$

Step 3.6

To write $3 v^{2}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$3 v^{2} \cdot \frac{10}{10} + \frac{- 3 v w \cdot 2 - v w \cdot 5}{10} + \frac{w^{2}}{10}$

Step 3.7

Combine $3 v^{2}$ and $\frac{10}{10}$ .

$\frac{3 v^{2} \cdot 10}{10} + \frac{- 3 v w \cdot 2 - v w \cdot 5}{10} + \frac{w^{2}}{10}$

Step 3.8

Combine the numerators over the common denominator.

$\frac{3 v^{2} \cdot 10 - 3 v w \cdot 2 - v w \cdot 5}{10} + \frac{w^{2}}{10}$

Step 3.9

Combine the numerators over the common denominator.

$\frac{3 v^{2} \cdot 10 - 3 v w \cdot 2 - v w \cdot 5 + w^{2}}{10}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $10$ by $3$ .

$\frac{30 v^{2} - 3 v w \cdot 2 - v w \cdot 5 + w^{2}}{10}$

Step 4.2

Multiply $2$ by $- 3$ .

$\frac{30 v^{2} - 6 v w - v w \cdot 5 + w^{2}}{10}$

Step 4.3

Multiply $5$ by $- 1$ .

$\frac{30 v^{2} - 6 v w - 5 v w + w^{2}}{10}$

Step 4.4

Subtract $5 v w$ from $- 6 v w$ .

$\frac{30 v^{2} - 11 v w + w^{2}}{10}$

Step 4.5

Factor by grouping.

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Step 4.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 30 \cdot 1 = 30$ and whose sum is $b = - 11$ .

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Step 4.5.1.1

Reorder terms.

$\frac{30 v^{2} + w^{2} - 11 v w}{10}$

Step 4.5.1.2

Reorder $w^{2}$ and $- 11 v w$ .

$\frac{30 v^{2} - 11 v w + w^{2}}{10}$

Step 4.5.1.3

Factor $- 11$ out of $- 11 v w$ .

$\frac{30 v^{2} - 11 (v w) + w^{2}}{10}$

Step 4.5.1.4

Rewrite $- 11$ as $- 5$ plus $- 6$

$\frac{30 v^{2} + (- 5 - 6) (v w) + w^{2}}{10}$

Step 4.5.1.5

Apply the distributive property.

$\frac{30 v^{2} - 5 (v w) - 6 (v w) + w^{2}}{10}$

Step 4.5.1.6

Move parentheses.

$\frac{30 v^{2} - 5 v w - 6 v w + w^{2}}{10}$

Step 4.5.2

Factor out the greatest common factor from each group.

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Step 4.5.2.1

Group the first two terms and the last two terms.

$\frac{(30 v^{2} - 5 v w) - 6 v w + w^{2}}{10}$

Step 4.5.2.2

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

$\frac{5 v (6 v - w) - w (6 v - w)}{10}$

Step 4.5.3

Factor the polynomial by factoring out the greatest common factor, $6 v - w$ .

$\frac{(6 v - w) (5 v - w)}{10}$