Algebra Examples

$a^{3} + 27 = (a + 3) (a^{2} + m a + 9)$

Step 1

Rewrite the equation as $(a + 3) (a^{2} + m a + 9) = a^{3} + 27$ .

$(a + 3) (a^{2} + m a + 9) = a^{3} + 27$

Step 2

Divide each term in $(a + 3) (a^{2} + m a + 9) = a^{3} + 27$ by $a + 3$ and simplify.

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Step 2.1

Divide each term in $(a + 3) (a^{2} + m a + 9) = a^{3} + 27$ by $a + 3$ .

$\frac{(a + 3) (a^{2} + m a + 9)}{a + 3} = \frac{a^{3}}{a + 3} + \frac{27}{a + 3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $a + 3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{(a + 3) (a^{2} + m a + 9)}{a + 3} = \frac{a^{3}}{a + 3} + \frac{27}{a + 3}$

Step 2.2.1.2

Divide $a^{2} + m a + 9$ by $1$ .

$a^{2} + m a + 9 = \frac{a^{3}}{a + 3} + \frac{27}{a + 3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Combine the numerators over the common denominator.

$a^{2} + m a + 9 = \frac{a^{3} + 27}{a + 3}$

Step 2.3.2

Simplify the numerator.

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Step 2.3.2.1

Rewrite $27$ as $3^{3}$ .

$a^{2} + m a + 9 = \frac{a^{3} + 3^{3}}{a + 3}$

Step 2.3.2.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = a$ and $b = 3$ .

$a^{2} + m a + 9 = \frac{(a + 3) (a^{2} - a \cdot 3 + 3^{2})}{a + 3}$

Step 2.3.2.3

Simplify.

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Step 2.3.2.3.1

Multiply $3$ by $- 1$ .

$a^{2} + m a + 9 = \frac{(a + 3) (a^{2} - 3 a + 3^{2})}{a + 3}$

Step 2.3.2.3.2

Raise $3$ to the power of $2$ .

$a^{2} + m a + 9 = \frac{(a + 3) (a^{2} - 3 a + 9)}{a + 3}$

Step 2.3.3

Cancel the common factor of $a + 3$ .

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Step 2.3.3.1

Cancel the common factor.

$a^{2} + m a + 9 = \frac{(a + 3) (a^{2} - 3 a + 9)}{a + 3}$

Step 2.3.3.2

Divide $a^{2} - 3 a + 9$ by $1$ .

$a^{2} + m a + 9 = a^{2} - 3 a + 9$

Step 3

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

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Step 3.1

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

$m a + 9 = a^{2} - 3 a + 9 - a^{2}$

Step 3.2

Subtract $9$ from both sides of the equation.

$m a = a^{2} - 3 a + 9 - a^{2} - 9$

Step 3.3

Combine the opposite terms in $a^{2} - 3 a + 9 - a^{2} - 9$ .

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Step 3.3.1

Subtract $a^{2}$ from $a^{2}$ .

$m a = - 3 a + 9 + 0 - 9$

Step 3.3.2

Add $- 3 a + 9$ and $0$ .

$m a = - 3 a + 9 - 9$

Step 3.3.3

Subtract $9$ from $9$ .

$m a = - 3 a + 0$

Step 3.3.4

Add $- 3 a$ and $0$ .

$m a = - 3 a$

Step 4

Divide each term in $m a = - 3 a$ by $a$ and simplify.

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Step 4.1

Divide each term in $m a = - 3 a$ by $a$ .

$\frac{m a}{a} = \frac{- 3 a}{a}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $a$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{m a}{a} = \frac{- 3 a}{a}$

Step 4.2.1.2

Divide $m$ by $1$ .

$m = \frac{- 3 a}{a}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $a$ .

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Step 4.3.1.1

Cancel the common factor.

$m = \frac{- 3 a}{a}$

Step 4.3.1.2

Divide $- 3$ by $1$ .

$m = - 3$