Identify the Zeros and Their Multiplicities x^4-4x^3+3x^2

$x^{4} - 4 x^{3} + 3 x^{2}$

Set $x^{4} - 4 x^{3} + 3 x^{2}$ equal to $0$ .

$x^{4} - 4 x^{3} + 3 x^{2} = 0$

Solve for $x$ .

Factor the left side of the equation.

Factor $x^{2}$ out of $x^{4} - 4 x^{3} + 3 x^{2}$ .

Factor $x^{2}$ out of $x^{4}$ .

$x^{2} x^{2} - 4 x^{3} + 3 x^{2} = 0$

Factor $x^{2}$ out of $- 4 x^{3}$ .

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

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

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

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

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

Factor $x^{2}$ out of $x^{2} (x^{2} - 4 x) + x^{2} \cdot 3$ .

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

Factor.

Factor $x^{2} - 4 x + 3$ 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 $3$ and whose sum is $- 4$ .

$- 3, - 1$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

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

$x^{2} = 0$

$x - 3 = 0$

$x - 1 = 0$

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

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

$x^{2} = 0$

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

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

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.

$x = 3$

Set $x - 1$ equal to $0$ and solve for $x$ .

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

$x = 0$ (Multiplicity of $2$ )

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

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

$x = 0$ (Multiplicity of $2$ )

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

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

Identify the Zeros and Their Multiplicities x^4-4x^3+3x^2

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