Algebra Examples

\frac{2 m^{6} + 6 m}{3 m^{14} + 12 m^{9} + 9 m^{4}}

$\frac{2 m^{6} + 6 m}{3 m^{14} + 12 m^{9} + 9 m^{4}}$

Step 1

Set the denominator in

\frac{2 m^{6} + 6 m}{3 m^{14} + 12 m^{9} + 9 m^{4}}

$\frac{2 m^{6} + 6 m}{3 m^{14} + 12 m^{9} + 9 m^{4}}$ equal to

0

$0$ to find where the expression is undefined.

3 m^{14} + 12 m^{9} + 9 m^{4} = 0

$3 m^{14} + 12 m^{9} + 9 m^{4} = 0$

Step 2

Solve for

m

$m$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor

3 m^{4}

$3 m^{4}$ out of

3 m^{14} + 12 m^{9} + 9 m^{4}

$3 m^{14} + 12 m^{9} + 9 m^{4}$ .

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Step 2.1.1.1

Factor

3 m^{4}

$3 m^{4}$ out of

3 m^{14}

$3 m^{14}$ .

3 m^{4} (m^{10}) + 12 m^{9} + 9 m^{4} = 0

$3 m^{4} (m^{10}) + 12 m^{9} + 9 m^{4} = 0$

Step 2.1.1.2

Factor

3 m^{4}

$3 m^{4}$ out of

12 m^{9}

$12 m^{9}$ .

3 m^{4} (m^{10}) + 3 m^{4} (4 m^{5}) + 9 m^{4} = 0

$3 m^{4} (m^{10}) + 3 m^{4} (4 m^{5}) + 9 m^{4} = 0$

Step 2.1.1.3

Factor

3 m^{4}

$3 m^{4}$ out of

9 m^{4}

$9 m^{4}$ .

3 m^{4} (m^{10}) + 3 m^{4} (4 m^{5}) + 3 m^{4} (3) = 0

$3 m^{4} (m^{10}) + 3 m^{4} (4 m^{5}) + 3 m^{4} (3) = 0$

Step 2.1.1.4

Factor

3 m^{4}

$3 m^{4}$ out of

3 m^{4} (m^{10}) + 3 m^{4} (4 m^{5})

$3 m^{4} (m^{10}) + 3 m^{4} (4 m^{5})$ .

3 m^{4} (m^{10} + 4 m^{5}) + 3 m^{4} (3) = 0

$3 m^{4} (m^{10} + 4 m^{5}) + 3 m^{4} (3) = 0$

Step 2.1.1.5

Factor

3 m^{4}

$3 m^{4}$ out of

3 m^{4} (m^{10} + 4 m^{5}) + 3 m^{4} (3)

$3 m^{4} (m^{10} + 4 m^{5}) + 3 m^{4} (3)$ .

3 m^{4} (m^{10} + 4 m^{5} + 3) = 0

$3 m^{4} (m^{10} + 4 m^{5} + 3) = 0$

3 m^{4} (m^{10} + 4 m^{5} + 3) = 0

$3 m^{4} (m^{10} + 4 m^{5} + 3) = 0$

Step 2.1.2

Rewrite

m^{10}

$m^{10}$ as

{(m^{5})}^{2}

${(m^{5})}^{2}$ .

3 m^{4} ({(m^{5})}^{2} + 4 m^{5} + 3) = 0

$3 m^{4} ({(m^{5})}^{2} + 4 m^{5} + 3) = 0$

Step 2.1.3

Let

u = m^{5}

$u = m^{5}$ . Substitute

u

$u$ for all occurrences of

m^{5}

$m^{5}$ .

3 m^{4} (u^{2} + 4 u + 3) = 0

$3 m^{4} (u^{2} + 4 u + 3) = 0$

Step 2.1.4

Factor

u^{2} + 4 u + 3

$u^{2} + 4 u + 3$ using the AC method.

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Step 2.1.4.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

3

$3$ and whose sum is

4

$4$ .

1, 3

$1, 3$

Step 2.1.4.2

Write the factored form using these integers.

3 m^{4} ((u + 1) (u + 3)) = 0

$3 m^{4} ((u + 1) (u + 3)) = 0$

3 m^{4} ((u + 1) (u + 3)) = 0

$3 m^{4} ((u + 1) (u + 3)) = 0$

Step 2.1.5

Factor.

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Step 2.1.5.1

Replace all occurrences of

u

$u$ with

m^{5}

$m^{5}$ .

3 m^{4} ((m^{5} + 1) (m^{5} + 3)) = 0

$3 m^{4} ((m^{5} + 1) (m^{5} + 3)) = 0$

Step 2.1.5.2

Remove unnecessary parentheses.

3 m^{4} (m^{5} + 1) (m^{5} + 3) = 0

$3 m^{4} (m^{5} + 1) (m^{5} + 3) = 0$

3 m^{4} (m^{5} + 1) (m^{5} + 3) = 0

$3 m^{4} (m^{5} + 1) (m^{5} + 3) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$m^{4} = 0$

$m^{5} + 1 = 0$

$m^{5} + 3 = 0$

Step 2.3

Set

$m^{4}$ equal to

$0$ and solve for

$m$ .

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Step 2.3.1

Set

$m^{4}$ equal to

$0$ .

$m^{4} = 0$

Step 2.3.2

Solve

$m^{4} = 0$ for

$m$ .

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Step 2.3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$m = \pm \sqrt[4]{0}$

Step 2.3.2.2

Simplify

$\pm \sqrt[4]{0}$ .

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Step 2.3.2.2.1

Rewrite

$0$ as

$0^{4}$ .

$m = \pm \sqrt[4]{0^{4}}$

Step 2.3.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$m = \pm 0$

Step 2.3.2.2.3

Plus or minus

$0$ is

$0$ .

$m = 0$

Step 2.4

Set

$m^{5} + 1$ equal to

$0$ and solve for

$m$ .

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Step 2.4.1

Set

$m^{5} + 1$ equal to

$0$ .

$m^{5} + 1 = 0$

Step 2.4.2

Solve

$m^{5} + 1 = 0$ for

$m$ .

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Step 2.4.2.1

Subtract

$1$ from both sides of the equation.

$m^{5} = - 1$

Step 2.4.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$m = \sqrt[5]{- 1}$

Step 2.4.2.3

Simplify

$\sqrt[5]{- 1}$ .

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Step 2.4.2.3.1

Rewrite

$- 1$ as

${(- 1)}^{5}$ .

$m = \sqrt[5]{{(- 1)}^{5}}$

Step 2.4.2.3.2

Pull terms out from under the radical, assuming real numbers.

$m = - 1$

Step 2.5

Set

$m^{5} + 3$ equal to

$0$ and solve for

$m$ .

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Step 2.5.1

Set

$m^{5} + 3$ equal to

$0$ .

$m^{5} + 3 = 0$

Step 2.5.2

Solve

$m^{5} + 3 = 0$ for

$m$ .

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Step 2.5.2.1

Subtract

$3$ from both sides of the equation.

$m^{5} = - 3$

Step 2.5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$m = \sqrt[5]{- 3}$

Step 2.5.2.3

Simplify

$\sqrt[5]{- 3}$ .

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Step 2.5.2.3.1

Rewrite

$- 3$ as

${(- 1)}^{5} \cdot 3$ .

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Step 2.5.2.3.1.1

Rewrite

$- 3$ as

$- 1 (3)$ .

$m = \sqrt[5]{- 1 (3)}$

Step 2.5.2.3.1.2

Rewrite

$- 1$ as

${(- 1)}^{5}$ .

$m = \sqrt[5]{{(- 1)}^{5} \cdot 3}$

Step 2.5.2.3.2

Pull terms out from under the radical.

$m = - 1 \sqrt[5]{3}$

Step 2.5.2.3.3

Rewrite

$- 1 \sqrt[5]{3}$ as

$- \sqrt[5]{3}$ .

$m = - \sqrt[5]{3}$

Step 2.6

The final solution is all the values that make

$3 m^{4} (m^{5} + 1) (m^{5} + 3) = 0$ true.

$m = 0, - 1, - \sqrt[5]{3}$

Step 3

The equation is undefined where the denominator equals

$0$ , the argument of a square root is less than

$0$ , or the argument of a logarithm is less than or equal to

$0$ .

$m = - \sqrt[5]{3}, m = - 1, m = 0$

Step 4