Identify the Zeros and Their Multiplicities f(x)=x^4+13x^2+36

$f (x) = x^{4} + 13 x^{2} + 36$

Step 1

Set $x^{4} + 13 x^{2} + 36$ equal to $0$ .

$x^{4} + 13 x^{2} + 36 = 0$

Step 2

Solve for $x$ .

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} + 13 u + 36 = 0$

$u = x^{2}$

Factor $u^{2} + 13 u + 36$ 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 $36$ and whose sum is $13$ .

$4, 9$

Write the factored form using these integers.

$(u + 4) (u + 9) = 0$

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

$u + 4 = 0$

$u + 9 = 0$

Set $u + 4$ equal to $0$ and solve for $u$ .

Set $u + 4$ equal to $0$ .

$u + 4 = 0$

Subtract $4$ from both sides of the equation.

$u = - 4$

Set $u + 9$ equal to $0$ and solve for $u$ .

Set $u + 9$ equal to $0$ .

$u + 9 = 0$

Subtract $9$ from both sides of the equation.

$u = - 9$

The final solution is all the values that make $(u + 4) (u + 9) = 0$ true. The multiplicity of a root is the number of times the root appears.

$u = - 4$ (Multiplicity of $1$ )

$u = - 9$ (Multiplicity of $1$ )

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = - 4$

${(x^{2})}^{1} = - 9$

Solve the first equation for $x$ .

$x^{2} = - 4$

Solve the equation for $x$ .

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

$x = \pm \sqrt{- 4}$

Simplify $\pm \sqrt{- 4}$ .

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{4}$

Rewrite $4$ as $2^{2}$ .

$x = \pm i \cdot \sqrt{2^{2}}$

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

$x = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$x = \pm 2 i$

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.

$x = 2 i$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 i$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 i, - 2 i$

The multiplicity of a root is the number of times the root appears. For example, a factor of ${(x + 5)}^{3}$ would have a root at $x = - 5$ with multiplicity of $3$ .

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

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

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

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

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 9$

Solve the equation for $x$ .

Remove parentheses.

$x^{2} = - 9$

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

$x = \pm \sqrt{- 9}$

Simplify $\pm \sqrt{- 9}$ .

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{9}$

Rewrite $9$ as $3^{2}$ .

$x = \pm i \cdot \sqrt{3^{2}}$

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

$x = \pm i \cdot 3$

Move $3$ to the left of $i$ .

$x = \pm 3 i$

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.

$x = 3 i$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3 i$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3 i, - 3 i$

The multiplicity of a root is the number of times the root appears. For example, a factor of ${(x + 5)}^{3}$ would have a root at $x = - 5$ with multiplicity of $3$ .

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

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

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

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

The solution to $x^{4} + 13 x^{2} + 36 = 0$ is $x = 2 i, - 2 i, 3 i, - 3 i$ .

$x = 2 i, - 2 i, 3 i, - 3 i$

Step 3

Identify the Zeros and Their Multiplicities f(x)=x^4+13x^2+36

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