Identify the Zeros and Their Multiplicities f(x)=3x^3-6x^2-15x+30

$f (x) = 3 x^{3} - 6 x^{2} - 15 x + 30$

Step 1

Set $3 x^{3} - 6 x^{2} - 15 x + 30$ equal to $0$ .

$3 x^{3} - 6 x^{2} - 15 x + 30 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $3$ out of $3 x^{3} - 6 x^{2} - 15 x + 30$ .

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

$3 (x^{3}) - 6 x^{2} - 15 x + 30 = 0$

Factor $3$ out of $- 6 x^{2}$ .

$3 (x^{3}) + 3 (- 2 x^{2}) - 15 x + 30 = 0$

Factor $3$ out of $- 15 x$ .

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

Factor $3$ out of $30$ .

$3 x^{3} + 3 (- 2 x^{2}) + 3 (- 5 x) + 3 \cdot 10 = 0$

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

$3 (x^{3} - 2 x^{2}) + 3 (- 5 x) + 3 \cdot 10 = 0$

Factor $3$ out of $3 (x^{3} - 2 x^{2}) + 3 (- 5 x)$ .

$3 (x^{3} - 2 x^{2} - 5 x) + 3 \cdot 10 = 0$

Factor $3$ out of $3 (x^{3} - 2 x^{2} - 5 x) + 3 \cdot 10$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

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

Factor.

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

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

Remove unnecessary parentheses.

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

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x^{2} - 5 = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set $x^{2} - 5$ equal to $0$ and solve for $x$ .

Set $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Solve $x^{2} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$x^{2} = 5$

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

$x = \pm \sqrt{5}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{5}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{5}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{5}, - \sqrt{5}$

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

$x = 2$ (Multiplicity of $1$ )

$x = \sqrt{5}$ (Multiplicity of $1$ )

$x = - \sqrt{5}$ (Multiplicity of $1$ )

$x = 2$ (Multiplicity of $1$ )

$x = \sqrt{5}$ (Multiplicity of $1$ )

$x = - \sqrt{5}$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities f(x)=3x^3-6x^2-15x+30

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