Find Where Undefined/Discontinuous (5x^8-20x^4)/(5x^8+20x^6+20x^4)

$\frac{5 x^{8} - 20 x^{4}}{5 x^{8} + 20 x^{6} + 20 x^{4}}$

Step 1

Set the denominator in $\frac{5 x^{8} - 20 x^{4}}{5 x^{8} + 20 x^{6} + 20 x^{4}}$ equal to $0$ to find where the expression is undefined.

$5 x^{8} + 20 x^{6} + 20 x^{4} = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $5 x^{4}$ out of $5 x^{8} + 20 x^{6} + 20 x^{4}$ .

Factor $5 x^{4}$ out of $5 x^{8}$ .

$5 x^{4} (x^{4}) + 20 x^{6} + 20 x^{4} = 0$

Factor $5 x^{4}$ out of $20 x^{6}$ .

$5 x^{4} (x^{4}) + 5 x^{4} (4 x^{2}) + 20 x^{4} = 0$

Factor $5 x^{4}$ out of $20 x^{4}$ .

$5 x^{4} (x^{4}) + 5 x^{4} (4 x^{2}) + 5 x^{4} (4) = 0$

Factor $5 x^{4}$ out of $5 x^{4} (x^{4}) + 5 x^{4} (4 x^{2})$ .

$5 x^{4} (x^{4} + 4 x^{2}) + 5 x^{4} (4) = 0$

Factor $5 x^{4}$ out of $5 x^{4} (x^{4} + 4 x^{2}) + 5 x^{4} (4)$ .

$5 x^{4} (x^{4} + 4 x^{2} + 4) = 0$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$5 x^{4} ({(x^{2})}^{2} + 4 x^{2} + 4) = 0$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$5 x^{4} (u^{2} + 4 u + 4) = 0$

Factor using the perfect square rule.

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

$5 x^{4} (u^{2} + 4 u + 2^{2}) = 0$

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

$4 u = 2 \cdot u \cdot 2$

Rewrite the polynomial.

$5 x^{4} (u^{2} + 2 \cdot u \cdot 2 + 2^{2}) = 0$

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

$5 x^{4} {(u + 2)}^{2} = 0$

Replace all occurrences of $u$ with $x^{2}$ .

$5 x^{4} {(x^{2} + 2)}^{2} = 0$

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

$x^{4} = 0$

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

Set $x^{4}$ equal to $0$ and solve for $x$ .

Set $x^{4}$ equal to $0$ .

$x^{4} = 0$

Solve $x^{4} = 0$ for $x$ .

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

$x = \pm \sqrt[4]{0}$

Simplify $\pm \sqrt[4]{0}$ .

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

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

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

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

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

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

$x^{2} + 2 = 0$

Solve for $x$ .

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

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

$x = \pm \sqrt{- 2}$

Simplify $\pm \sqrt{- 2}$ .

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

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

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

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.

$x = i \sqrt{2}$

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

$x = - i \sqrt{2}$

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

$x = i \sqrt{2}, - i \sqrt{2}$

The final solution is all the values that make $5 x^{4} {(x^{2} + 2)}^{2} = 0$ true.

$x = 0, i \sqrt{2}, - i \sqrt{2}$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = 0$

Step 4

Find Where Undefined/Discontinuous (5x^8-20x^4)/(5x^8+20x^6+20x^4)

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