Find the Domain x/(x^3+343)

$\frac{x}{x^{3} + 343}$

Step 1

Set the denominator in $\frac{x}{x^{3} + 343}$ equal to $0$ to find where the expression is undefined.

$x^{3} + 343 = 0$

Step 2

Solve for $x$ .

Subtract $343$ from both sides of the equation.

$x^{3} = - 343$

Add $343$ to both sides of the equation.

$x^{3} + 343 = 0$

Factor the left side of the equation.

Rewrite $343$ as $7^{3}$ .

$x^{3} + 7^{3} = 0$

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 7$ .

$(x + 7) (x^{2} - x \cdot 7 + 7^{2}) = 0$

Simplify.

Multiply $7$ by $- 1$ .

$(x + 7) (x^{2} - 7 x + 7^{2}) = 0$

Raise $7$ to the power of $2$ .

$(x + 7) (x^{2} - 7 x + 49) = 0$

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

$x + 7 = 0$

$x^{2} - 7 x + 49 = 0$

Set $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

Set $x^{2} - 7 x + 49$ equal to $0$ and solve for $x$ .

Set $x^{2} - 7 x + 49$ equal to $0$ .

$x^{2} - 7 x + 49 = 0$

Solve $x^{2} - 7 x + 49 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 7$ , and $c = 49$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (1 \cdot 49)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 1 \cdot 49}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 49$ .

Multiply $- 4$ by $1$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 49}}{2 \cdot 1}$

Multiply $- 4$ by $49$ .

$x = \frac{7 \pm \sqrt{49 - 196}}{2 \cdot 1}$

Subtract $196$ from $49$ .

$x = \frac{7 \pm \sqrt{- 147}}{2 \cdot 1}$

Rewrite $- 147$ as $- 1 (147)$ .

$x = \frac{7 \pm \sqrt{- 1 \cdot 147}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (147)}$ as $\sqrt{- 1} \cdot \sqrt{147}$ .

$x = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{147}}{2 \cdot 1}$

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

$x = \frac{7 \pm i \cdot \sqrt{147}}{2 \cdot 1}$

Rewrite $147$ as $7^{2} \cdot 3$ .

Factor $49$ out of $147$ .

$x = \frac{7 \pm i \cdot \sqrt{49 (3)}}{2 \cdot 1}$

Rewrite $49$ as $7^{2}$ .

$x = \frac{7 \pm i \cdot \sqrt{7^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{7 \pm i \cdot (7 \sqrt{3})}{2 \cdot 1}$

Move $7$ to the left of $i$ .

$x = \frac{7 \pm 7 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{7 \pm 7 i \sqrt{3}}{2}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 1 \cdot 49}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 49$ .

Multiply $- 4$ by $1$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 49}}{2 \cdot 1}$

Multiply $- 4$ by $49$ .

$x = \frac{7 \pm \sqrt{49 - 196}}{2 \cdot 1}$

Subtract $196$ from $49$ .

$x = \frac{7 \pm \sqrt{- 147}}{2 \cdot 1}$

Rewrite $- 147$ as $- 1 (147)$ .

$x = \frac{7 \pm \sqrt{- 1 \cdot 147}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (147)}$ as $\sqrt{- 1} \cdot \sqrt{147}$ .

$x = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{147}}{2 \cdot 1}$

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

$x = \frac{7 \pm i \cdot \sqrt{147}}{2 \cdot 1}$

Rewrite $147$ as $7^{2} \cdot 3$ .

Factor $49$ out of $147$ .

$x = \frac{7 \pm i \cdot \sqrt{49 (3)}}{2 \cdot 1}$

Rewrite $49$ as $7^{2}$ .

$x = \frac{7 \pm i \cdot \sqrt{7^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{7 \pm i \cdot (7 \sqrt{3})}{2 \cdot 1}$

Move $7$ to the left of $i$ .

$x = \frac{7 \pm 7 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{7 \pm 7 i \sqrt{3}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{7 + 7 i \sqrt{3}}{2}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 1 \cdot 49}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 49$ .

Multiply $- 4$ by $1$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 49}}{2 \cdot 1}$

Multiply $- 4$ by $49$ .

$x = \frac{7 \pm \sqrt{49 - 196}}{2 \cdot 1}$

Subtract $196$ from $49$ .

$x = \frac{7 \pm \sqrt{- 147}}{2 \cdot 1}$

Rewrite $- 147$ as $- 1 (147)$ .

$x = \frac{7 \pm \sqrt{- 1 \cdot 147}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (147)}$ as $\sqrt{- 1} \cdot \sqrt{147}$ .

$x = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{147}}{2 \cdot 1}$

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

$x = \frac{7 \pm i \cdot \sqrt{147}}{2 \cdot 1}$

Rewrite $147$ as $7^{2} \cdot 3$ .

Factor $49$ out of $147$ .

$x = \frac{7 \pm i \cdot \sqrt{49 (3)}}{2 \cdot 1}$

Rewrite $49$ as $7^{2}$ .

$x = \frac{7 \pm i \cdot \sqrt{7^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{7 \pm i \cdot (7 \sqrt{3})}{2 \cdot 1}$

Move $7$ to the left of $i$ .

$x = \frac{7 \pm 7 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{7 \pm 7 i \sqrt{3}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{7 - 7 i \sqrt{3}}{2}$

The final answer is the combination of both solutions.

$x = \frac{7 + 7 i \sqrt{3}}{2}, \frac{7 - 7 i \sqrt{3}}{2}$

The final solution is all the values that make $(x + 7) (x^{2} - 7 x + 49) = 0$ true.

$x = - 7, \frac{7 + 7 i \sqrt{3}}{2}, \frac{7 - 7 i \sqrt{3}}{2}$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 7) \cup (- 7, \infty)$

Set-Builder Notation:

${x | x \neq - 7}$

Step 4

Find the Domain x/(x^3+343)

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