Basic Math Examples

$(b - c) c^{2} + (c - a) c + (a - b) = 0$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$b c^{2} - c \cdot c^{2} + (c - a) c + a - b = 0$

Step 1.2

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

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Step 1.2.1

Move $c^{2}$ .

$b c^{2} - (c^{2} c) + (c - a) c + a - b = 0$

Step 1.2.2

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

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Step 1.2.2.1

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

$b c^{2} - (c^{2} c^{1}) + (c - a) c + a - b = 0$

Step 1.2.2.2

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

$b c^{2} - c^{2 + 1} + (c - a) c + a - b = 0$

Step 1.2.3

Add $2$ and $1$ .

$b c^{2} - c^{3} + (c - a) c + a - b = 0$

Step 1.3

Apply the distributive property.

$b c^{2} - c^{3} + c \cdot c - a c + a - b = 0$

Step 1.4

Multiply $c$ by $c$ .

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

Step 2

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

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Step 2.1

Add $c^{3}$ to both sides of the equation.

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

Step 2.2

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

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

Step 2.3

Add $a c$ to both sides of the equation.

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

Step 2.4

Subtract $a$ from both sides of the equation.

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

Step 3

Factor $b$ out of $b c^{2} - b$ .

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Step 3.1

Factor $b$ out of $b c^{2}$ .

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

Step 3.2

Factor $b$ out of $- b$ .

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

Step 3.3

Factor $b$ out of $b (c^{2}) + b \cdot - 1$ .

$b (c^{2} - 1) = c^{3} - c^{2} + a c - a$

Step 4

Rewrite $1$ as $1^{2}$ .

$b (c^{2} - 1^{2}) = c^{3} - c^{2} + a c - a$

Step 5

Factor.

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Step 5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = c$ and $b = 1$ .

$b ((c + 1) (c - 1)) = c^{3} - c^{2} + a c - a$

Step 5.2

Remove unnecessary parentheses.

$b (c + 1) (c - 1) = c^{3} - c^{2} + a c - a$

Step 6

Divide each term in $b (c + 1) (c - 1) = c^{3} - c^{2} + a c - a$ by $(c + 1) (c - 1)$ and simplify.

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Step 6.1

Divide each term in $b (c + 1) (c - 1) = c^{3} - c^{2} + a c - a$ by $(c + 1) (c - 1)$ .

$\frac{b (c + 1) (c - 1)}{(c + 1) (c - 1)} = \frac{c^{3}}{(c + 1) (c - 1)} + \frac{- c^{2}}{(c + 1) (c - 1)} + \frac{a c}{(c + 1) (c - 1)} + \frac{- a}{(c + 1) (c - 1)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $c + 1$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{b (c + 1) (c - 1)}{(c + 1) (c - 1)} = \frac{c^{3}}{(c + 1) (c - 1)} + \frac{- c^{2}}{(c + 1) (c - 1)} + \frac{a c}{(c + 1) (c - 1)} + \frac{- a}{(c + 1) (c - 1)}$

Step 6.2.1.2

Rewrite the expression.

$\frac{b (c - 1)}{c - 1} = \frac{c^{3}}{(c + 1) (c - 1)} + \frac{- c^{2}}{(c + 1) (c - 1)} + \frac{a c}{(c + 1) (c - 1)} + \frac{- a}{(c + 1) (c - 1)}$

Step 6.2.2

Cancel the common factor of $c - 1$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{b (c - 1)}{c - 1} = \frac{c^{3}}{(c + 1) (c - 1)} + \frac{- c^{2}}{(c + 1) (c - 1)} + \frac{a c}{(c + 1) (c - 1)} + \frac{- a}{(c + 1) (c - 1)}$

Step 6.2.2.2

Divide $b$ by $1$ .

$b = \frac{c^{3}}{(c + 1) (c - 1)} + \frac{- c^{2}}{(c + 1) (c - 1)} + \frac{a c}{(c + 1) (c - 1)} + \frac{- a}{(c + 1) (c - 1)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Move the negative in front of the fraction.

$b = \frac{c^{3}}{(c + 1) (c - 1)} - \frac{c^{2}}{(c + 1) (c - 1)} + \frac{a c}{(c + 1) (c - 1)} + \frac{- a}{(c + 1) (c - 1)}$

Step 6.3.1.2

Move the negative in front of the fraction.

$b = \frac{c^{3}}{(c + 1) (c - 1)} - \frac{c^{2}}{(c + 1) (c - 1)} + \frac{a c}{(c + 1) (c - 1)} - \frac{a}{(c + 1) (c - 1)}$