Identify the Zeros and Their Multiplicities f(x)=x^5+10x^4+25x^3

$f (x) = x^{5} + 10 x^{4} + 25 x^{3}$

Step 1

Set $x^{5} + 10 x^{4} + 25 x^{3}$ equal to $0$ .

$x^{5} + 10 x^{4} + 25 x^{3} = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $x^{3}$ out of $x^{5} + 10 x^{4} + 25 x^{3}$ .

Factor $x^{3}$ out of $x^{5}$ .

$x^{3} x^{2} + 10 x^{4} + 25 x^{3} = 0$

Factor $x^{3}$ out of $10 x^{4}$ .

$x^{3} x^{2} + x^{3} (10 x) + 25 x^{3} = 0$

Factor $x^{3}$ out of $25 x^{3}$ .

$x^{3} x^{2} + x^{3} (10 x) + x^{3} \cdot 25 = 0$

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (10 x)$ .

$x^{3} (x^{2} + 10 x) + x^{3} \cdot 25 = 0$

Factor $x^{3}$ out of $x^{3} (x^{2} + 10 x) + x^{3} \cdot 25$ .

$x^{3} (x^{2} + 10 x + 25) = 0$

Factor using the perfect square rule.

Rewrite $25$ as $5^{2}$ .

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

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

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

$x^{3} = 0$

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

Set $x^{3}$ equal to $0$ and solve for $x$ .

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Solve $x^{3} = 0$ for $x$ .

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

$x = \sqrt[3]{0}$

Simplify $\sqrt[3]{0}$ .

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

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

$x = 0$

Set ${(x + 5)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x + 5)}^{2}$ equal to $0$ .

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

Solve ${(x + 5)}^{2} = 0$ for $x$ .

Set the $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

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

$x = 0$ (Multiplicity of $3$ )

$x = - 5$ (Multiplicity of $2$ )

$x = 0$ (Multiplicity of $3$ )

$x = - 5$ (Multiplicity of $2$ )

Step 3

Identify the Zeros and Their Multiplicities f(x)=x^5+10x^4+25x^3

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