Identify the Zeros and Their Multiplicities y=(x^2-5)(x-1)^2(x-2)^3

$y = (x^{2} - 5) {(x - 1)}^{2} {(x - 2)}^{3}$

Set $(x^{2} - 5) {(x - 1)}^{2} {(x - 2)}^{3}$ equal to $0$ .

$(x^{2} - 5) {(x - 1)}^{2} {(x - 2)}^{3} = 0$

Solve for $x$ .

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

$x^{2} - 5 = 0$

${(x - 1)}^{2} = 0$

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

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 square 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.

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

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 1)}^{2}$ equal to $0$ .

${(x - 1)}^{2} = 0$

Solve ${(x - 1)}^{2} = 0$ for $x$ .

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set ${(x - 2)}^{3}$ equal to $0$ and solve for $x$ .

Set ${(x - 2)}^{3}$ equal to $0$ .

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

Solve ${(x - 2)}^{3} = 0$ for $x$ .

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x^{2} - 5) {(x - 1)}^{2} {(x - 2)}^{3} = 0$ true. The multiplicity of a root is the number of times the root appears.

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

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

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

$x = 2$ (Multiplicity of $3$ )

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

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

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

$x = 2$ (Multiplicity of $3$ )

Identify the Zeros and Their Multiplicities y=(x^2-5)(x-1)^2(x-2)^3

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