Identify the Zeros and Their Multiplicities f(x)=3x^3-21x^2-54x

$f (x) = 3 x^{3} - 21 x^{2} - 54 x$

Step 1

Set $3 x^{3} - 21 x^{2} - 54 x$ equal to $0$ .

$3 x^{3} - 21 x^{2} - 54 x = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $3 x$ out of $3 x^{3} - 21 x^{2} - 54 x$ .

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

$3 x (x^{2}) - 21 x^{2} - 54 x = 0$

Factor $3 x$ out of $- 21 x^{2}$ .

$3 x (x^{2}) + 3 x (- 7 x) - 54 x = 0$

Factor $3 x$ out of $- 54 x$ .

$3 x (x^{2}) + 3 x (- 7 x) + 3 x (- 18) = 0$

Factor $3 x$ out of $3 x (x^{2}) + 3 x (- 7 x)$ .

$3 x (x^{2} - 7 x) + 3 x (- 18) = 0$

Factor $3 x$ out of $3 x (x^{2} - 7 x) + 3 x (- 18)$ .

$3 x (x^{2} - 7 x - 18) = 0$

Factor.

Factor $x^{2} - 7 x - 18$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $- 7$ .

$- 9, 2$

Write the factored form using these integers.

$3 x ((x - 9) (x + 2)) = 0$

Remove unnecessary parentheses.

$3 x (x - 9) (x + 2) = 0$

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

$x = 0$

$x - 9 = 0$

$x + 2 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $3 x (x - 9) (x + 2) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 0$ (Multiplicity of $1$ )

$x = 9$ (Multiplicity of $1$ )

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

$x = 0$ (Multiplicity of $1$ )

$x = 9$ (Multiplicity of $1$ )

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

Step 3

Identify the Zeros and Their Multiplicities f(x)=3x^3-21x^2-54x

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