Identify the Zeros and Their Multiplicities (x-3)(5x-6)(x-6)^3=0

$(x - 3) (5 x - 6) {(x - 6)}^{3} = 0$

Step 1

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

$x - 3 = 0$

$5 x - 6 = 0$

${(x - 6)}^{3} = 0$

Step 2

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 3

Set $5 x - 6$ equal to $0$ and solve for $x$ .

Set $5 x - 6$ equal to $0$ .

$5 x - 6 = 0$

Solve $5 x - 6 = 0$ for $x$ .

Add $6$ to both sides of the equation.

$5 x = 6$

Divide each term in $5 x = 6$ by $5$ and simplify.

Divide each term in $5 x = 6$ by $5$ .

$\frac{5 x}{5} = \frac{6}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{6}{5}$

Divide $x$ by $1$ .

$x = \frac{6}{5}$

Step 4

Set ${(x - 6)}^{3}$ equal to $0$ and solve for $x$ .

Set ${(x - 6)}^{3}$ equal to $0$ .

${(x - 6)}^{3} = 0$

Solve ${(x - 6)}^{3} = 0$ for $x$ .

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Step 5

The final solution is all the values that make $(x - 3) (5 x - 6) {(x - 6)}^{3} = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 3$ (Multiplicity of $1$ )

$x = \frac{6}{5}$ (Multiplicity of $1$ )

$x = 6$ (Multiplicity of $3$ )

Step 6

Identify the Zeros and Their Multiplicities (x-3)(5x-6)(x-6)^3=0

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