Identify the Zeros and Their Multiplicities f(x)=x^4+55x^2-576

$f (x) = x^{4} + 55 x^{2} - 576$

Step 1

Set $x^{4} + 55 x^{2} - 576$ equal to $0$ .

$x^{4} + 55 x^{2} - 576 = 0$

Step 2

Solve for $x$ .

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

$u^{2} + 55 u - 576 = 0$

$u = x^{2}$

Factor $u^{2} + 55 u - 576$ 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 $- 576$ and whose sum is $55$ .

$- 9, 64$

Write the factored form using these integers.

$(u - 9) (u + 64) = 0$

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

$u - 9 = 0$

$u + 64 = 0$

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

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

$u - 9 = 0$

Add $9$ to both sides of the equation.

$u = 9$

Set $u + 64$ equal to $0$ and solve for $u$ .

Set $u + 64$ equal to $0$ .

$u + 64 = 0$

Subtract $64$ from both sides of the equation.

$u = - 64$

The final solution is all the values that make $(u - 9) (u + 64) = 0$ true. The multiplicity of a root is the number of times the root appears.

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

$u = - 64$ (Multiplicity of $1$ )

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

$x^{2} = 9$

${(x^{2})}^{1} = - 64$

Solve the first equation for $x$ .

$x^{2} = 9$

Solve the equation for $x$ .

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

$x = \pm \sqrt{9}$

Simplify $\pm \sqrt{9}$ .

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

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

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

$x = \pm 3$

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.

$x = 3$

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

$x = - 3$

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

$x = 3, - 3$

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 $3$ .

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

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

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

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

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 64$

Solve the equation for $x$ .

Remove parentheses.

$x^{2} = - 64$

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

$x = \pm \sqrt{- 64}$

Simplify $\pm \sqrt{- 64}$ .

Rewrite $- 64$ as $- 1 (64)$ .

$x = \pm \sqrt{- 1 (64)}$

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{64}$

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

$x = \pm i \cdot \sqrt{64}$

Rewrite $64$ as $8^{2}$ .

$x = \pm i \cdot \sqrt{8^{2}}$

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

$x = \pm i \cdot 8$

Move $8$ to the left of $i$ .

$x = \pm 8 i$

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.

$x = 8 i$

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

$x = - 8 i$

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

$x = 8 i, - 8 i$

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 $3$ .

$x = 8 i$ (Multiplicity of $1$ )

$x = - 8 i$ (Multiplicity of $1$ )

$x = 8 i$ (Multiplicity of $1$ )

$x = - 8 i$ (Multiplicity of $1$ )

The solution to $x^{4} + 55 x^{2} - 576 = 0$ is $x = 3, - 3, 8 i, - 8 i$ .

$x = 3, - 3, 8 i, - 8 i$

Step 3

Identify the Zeros and Their Multiplicities f(x)=x^4+55x^2-576

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