Find All Complex Solutions x^5-7x^4-30x^3=0

$x^{5} - 7 x^{4} - 30 x^{3} = 0$

Step 1

Factor the left side of the equation.

Factor $x^{3}$ out of $x^{5} - 7 x^{4} - 30 x^{3}$ .

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

$x^{3} x^{2} - 7 x^{4} - 30 x^{3} = 0$

Factor $x^{3}$ out of $- 7 x^{4}$ .

$x^{3} x^{2} + x^{3} (- 7 x) - 30 x^{3} = 0$

Factor $x^{3}$ out of $- 30 x^{3}$ .

$x^{3} x^{2} + x^{3} (- 7 x) + x^{3} \cdot - 30 = 0$

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (- 7 x)$ .

$x^{3} (x^{2} - 7 x) + x^{3} \cdot - 30 = 0$

Factor $x^{3}$ out of $x^{3} (x^{2} - 7 x) + x^{3} \cdot - 30$ .

$x^{3} (x^{2} - 7 x - 30) = 0$

Factor.

Factor $x^{2} - 7 x - 30$ 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 $- 30$ and whose sum is $- 7$ .

$- 10, 3$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

Step 2

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

$x^{3} = 0$

$x - 10 = 0$

$x + 3 = 0$

Step 3

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.

$x = 0$

Step 4

Set $x - 10$ equal to $0$ and solve for $x$ .

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Add $10$ to both sides of the equation.

$x = 10$

Step 5

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6

The final solution is all the values that make $x^{3} (x - 10) (x + 3) = 0$ true.

$x = 0, 10, - 3$

Find All Complex Solutions x^5-7x^4-30x^3=0

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