Basic Math Examples

$(v^{2} + a v - b v - c v) (v - b)$

Step 1

Expand $(v^{2} + a v - b v - c v) (v - b)$ by multiplying each term in the first expression by each term in the second expression.

$v^{2} v + v^{2} (- b) + a v \cdot v + a v (- b) - b v \cdot v - b v (- b) - c v \cdot v - c v (- b)$

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

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

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Step 2.1.1.1

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

$v^{2} v^{1} + v^{2} (- b) + a v \cdot v + a v (- b) - b v \cdot v - b v (- b) - c v \cdot v - c v (- b)$

Step 2.1.1.2

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

$v^{2 + 1} + v^{2} (- b) + a v \cdot v + a v (- b) - b v \cdot v - b v (- b) - c v \cdot v - c v (- b)$

Step 2.1.2

Add $2$ and $1$ .

$v^{3} + v^{2} (- b) + a v \cdot v + a v (- b) - b v \cdot v - b v (- b) - c v \cdot v - c v (- b)$

Step 2.2

Rewrite using the commutative property of multiplication.

$v^{3} - v^{2} b + a v \cdot v + a v (- b) - b v \cdot v - b v (- b) - c v \cdot v - c v (- b)$

Step 2.3

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

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Step 2.3.1

Move $v$ .

$v^{3} - v^{2} b + a (v \cdot v) + a v (- b) - b v \cdot v - b v (- b) - c v \cdot v - c v (- b)$

Step 2.3.2

Multiply $v$ by $v$ .

$v^{3} - v^{2} b + a v^{2} + a v (- b) - b v \cdot v - b v (- b) - c v \cdot v - c v (- b)$

Step 2.4

Rewrite using the commutative property of multiplication.

$v^{3} - v^{2} b + a v^{2} - a v b - b v \cdot v - b v (- b) - c v \cdot v - c v (- b)$

Step 2.5

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

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Step 2.5.1

Move $v$ .

$v^{3} - v^{2} b + a v^{2} - a v b - b (v \cdot v) - b v (- b) - c v \cdot v - c v (- b)$

Step 2.5.2

Multiply $v$ by $v$ .

$v^{3} - v^{2} b + a v^{2} - a v b - b v^{2} - b v (- b) - c v \cdot v - c v (- b)$

Step 2.6

Multiply $b$ by $b$ by adding the exponents.

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Step 2.6.1

Move $b$ .

$v^{3} - v^{2} b + a v^{2} - a v b - b v^{2} - (b \cdot b) v \cdot - 1 - c v \cdot v - c v (- b)$

Step 2.6.2

Multiply $b$ by $b$ .

$v^{3} - v^{2} b + a v^{2} - a v b - b v^{2} - b^{2} v \cdot - 1 - c v \cdot v - c v (- b)$

Step 2.7

Multiply $- b^{2} v \cdot - 1$ .

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Step 2.7.1

Multiply $- 1$ by $- 1$ .

$v^{3} - v^{2} b + a v^{2} - a v b - b v^{2} + 1 b^{2} v - c v \cdot v - c v (- b)$

Step 2.7.2

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

$v^{3} - v^{2} b + a v^{2} - a v b - b v^{2} + b^{2} v - c v \cdot v - c v (- b)$

Step 2.8

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

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Step 2.8.1

Move $v$ .

$v^{3} - v^{2} b + a v^{2} - a v b - b v^{2} + b^{2} v - c (v \cdot v) - c v (- b)$

Step 2.8.2

Multiply $v$ by $v$ .

$v^{3} - v^{2} b + a v^{2} - a v b - b v^{2} + b^{2} v - c v^{2} - c v (- b)$

Step 2.9

Multiply $- c v (- b)$ .

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Step 2.9.1

Multiply $- 1$ by $- 1$ .

$v^{3} - v^{2} b + a v^{2} - a v b - b v^{2} + b^{2} v - c v^{2} + 1 c v b$

Step 2.9.2

Multiply $c$ by $1$ .

$v^{3} - v^{2} b + a v^{2} - a v b - b v^{2} + b^{2} v - c v^{2} + c v b$

Step 3

Subtract $b v^{2}$ from $- v^{2} b$ .

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Step 3.1

Move $v^{2}$ .

$v^{3} - b v^{2} - b v^{2} + a v^{2} - a v b + b^{2} v - c v^{2} + c v b$

Step 3.2

Subtract $b v^{2}$ from $- b v^{2}$ .

$v^{3} - 2 b v^{2} + a v^{2} - a v b + b^{2} v - c v^{2} + c v b$