Find the Roots (Zeros) 3x^4-9x^3+x^2-3x

$3 x^{4} - 9 x^{3} + x^{2} - 3 x$

Set $3 x^{4} - 9 x^{3} + x^{2} - 3 x$ equal to $0$ .

$3 x^{4} - 9 x^{3} + x^{2} - 3 x = 0$

Solve for $x$ .

Factor the left side of the equation.

Factor $x$ out of $3 x^{4} - 9 x^{3} + x^{2} - 3 x$ .

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

$x (3 x^{3}) - 9 x^{3} + x^{2} - 3 x = 0$

Factor $x$ out of $- 9 x^{3}$ .

$x (3 x^{3}) + x (- 9 x^{2}) + x^{2} - 3 x = 0$

Factor $x$ out of $x^{2}$ .

$x (3 x^{3}) + x (- 9 x^{2}) + x \cdot x - 3 x = 0$

Factor $x$ out of $- 3 x$ .

$x (3 x^{3}) + x (- 9 x^{2}) + x \cdot x + x \cdot - 3 = 0$

Factor $x$ out of $x (3 x^{3}) + x (- 9 x^{2})$ .

$x (3 x^{3} - 9 x^{2}) + x \cdot x + x \cdot - 3 = 0$

Factor $x$ out of $x (3 x^{3} - 9 x^{2}) + x \cdot x$ .

$x (3 x^{3} - 9 x^{2} + x) + x \cdot - 3 = 0$

Factor $x$ out of $x (3 x^{3} - 9 x^{2} + x) + x \cdot - 3$ .

$x (3 x^{3} - 9 x^{2} + x - 3) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x ((3 x^{3} - 9 x^{2}) + x - 3) = 0$

Factor out the greatest common factor (GCF) from each group.

$x (3 x^{2} (x - 3) + 1 (x - 3)) = 0$

Factor.

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$x ((x - 3) (3 x^{2} + 1)) = 0$

Remove unnecessary parentheses.

$x (x - 3) (3 x^{2} + 1) = 0$

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

$x = 0$

$x - 3 = 0$

$3 x^{2} + 1 = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $3 x^{2} + 1$ equal to $0$ and solve for $x$ .

Set $3 x^{2} + 1$ equal to $0$ .

$3 x^{2} + 1 = 0$

Solve $3 x^{2} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$3 x^{2} = - 1$

Divide each term in $3 x^{2} = - 1$ by $3$ and simplify.

Divide each term in $3 x^{2} = - 1$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 1}{3}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 1}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{3}$

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

$x = \pm \sqrt{- \frac{1}{3}}$

Simplify $\pm \sqrt{- \frac{1}{3}}$ .

Rewrite $- \frac{1}{3}$ as $i^{2} \frac{1^{2}}{3}$ .

Rewrite $- 1$ as $i^{2}$ .

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

Rewrite $1$ as $1^{2}$ .

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

Pull terms out from under the radical.

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

One to any power is one.

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

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Any root of $1$ is $1$ .

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

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm i \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

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

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.

$x = \frac{i \sqrt{3}}{3}$

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

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

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

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

The final solution is all the values that make $x (x - 3) (3 x^{2} + 1) = 0$ true.

$x = 0, 3, \frac{i \sqrt{3}}{3}, - \frac{i \sqrt{3}}{3}$

Find the Roots (Zeros) 3x^4-9x^3+x^2-3x

Related Questions

Support LCMGCF.com by disabling your adblocker.