Algebra Examples

$f (x) = 3 x^{6} + 30 x^{5} + 75 x^{4}$

Step 1

Set $3 x^{6} + 30 x^{5} + 75 x^{4}$ equal to $0$ .

$3 x^{6} + 30 x^{5} + 75 x^{4} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $3 x^{4}$ out of $3 x^{6} + 30 x^{5} + 75 x^{4}$ .

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Step 2.1.1.1

Factor $3 x^{4}$ out of $3 x^{6}$ .

$3 x^{4} (x^{2}) + 30 x^{5} + 75 x^{4} = 0$

Step 2.1.1.2

Factor $3 x^{4}$ out of $30 x^{5}$ .

$3 x^{4} (x^{2}) + 3 x^{4} (10 x) + 75 x^{4} = 0$

Step 2.1.1.3

Factor $3 x^{4}$ out of $75 x^{4}$ .

$3 x^{4} (x^{2}) + 3 x^{4} (10 x) + 3 x^{4} (25) = 0$

Step 2.1.1.4

Factor $3 x^{4}$ out of $3 x^{4} (x^{2}) + 3 x^{4} (10 x)$ .

$3 x^{4} (x^{2} + 10 x) + 3 x^{4} (25) = 0$

Step 2.1.1.5

Factor $3 x^{4}$ out of $3 x^{4} (x^{2} + 10 x) + 3 x^{4} (25)$ .

$3 x^{4} (x^{2} + 10 x + 25) = 0$

Step 2.1.2

Factor using the perfect square rule.

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Step 2.1.2.1

Rewrite $25$ as $5^{2}$ .

$3 x^{4} (x^{2} + 10 x + 5^{2}) = 0$

Step 2.1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 2.1.2.3

Rewrite the polynomial.

$3 x^{4} (x^{2} + 2 \cdot x \cdot 5 + 5^{2}) = 0$

Step 2.1.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

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

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 $0$ .

$x^{4} = 0$

${(x + 5)}^{2} = 0$

Step 2.3

Set $x^{4}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{4}$ equal to $0$ .

$x^{4} = 0$

Step 2.3.2

Solve $x^{4} = 0$ for $x$ .

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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.

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

Step 2.3.2.2

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

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Step 2.3.2.2.1

Rewrite $0$ as $0^{4}$ .

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

Step 2.3.2.2.2

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

$x = \pm 0$

Step 2.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.4

Set ${(x + 5)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set ${(x + 5)}^{2}$ equal to $0$ .

${(x + 5)}^{2} = 0$

Step 2.4.2

Solve ${(x + 5)}^{2} = 0$ for $x$ .

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Step 2.4.2.1

Set the $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 2.4.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 2.5

The final solution is all the values that make $3 x^{4} {(x + 5)}^{2} = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 0$ (Multiplicity of $4$ )

$x = - 5$ (Multiplicity of $2$ )

$x = 0$ (Multiplicity of $4$ )

$x = - 5$ (Multiplicity of $2$ )

Step 3