Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^3+5x^2+75x+125

$f (x) = 3 x^{3} + 5 x^{2} + 75 x + 125$

Step 1

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{3} + 5 x^{2}) + 75 x + 125 = 0$

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

$x^{2} (3 x + 5) + 25 (3 x + 5) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 5$ .

$(3 x + 5) (x^{2} + 25) = 0$

Step 2

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

$3 x + 5 = 0$

$x^{2} + 25 = 0$

Step 3

Set $3 x + 5$ equal to $0$ and solve for $x$ .

Set $3 x + 5$ equal to $0$ .

$3 x + 5 = 0$

Solve $3 x + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Divide each term in $3 x = - 5$ by $3$ and simplify.

Divide each term in $3 x = - 5$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5}{3}$

Divide $x$ by $1$ .

$x = \frac{- 5}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{3}$

Step 4

Set $x^{2} + 25$ equal to $0$ and solve for $x$ .

Set $x^{2} + 25$ equal to $0$ .

$x^{2} + 25 = 0$

Solve $x^{2} + 25 = 0$ for $x$ .

Subtract $25$ from both sides of the equation.

$x^{2} = - 25$

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

$x = \pm \sqrt{- 25}$

Simplify $\pm \sqrt{- 25}$ .

Rewrite $- 25$ as $- 1 (25)$ .

$x = \pm \sqrt{- 1 (25)}$

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{25}$

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

$x = \pm i \cdot \sqrt{25}$

Rewrite $25$ as $5^{2}$ .

$x = \pm i \cdot \sqrt{5^{2}}$

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

$x = \pm i \cdot 5$

Move $5$ to the left of $i$ .

$x = \pm 5 i$

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.

$x = 5 i$

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

$x = - 5 i$

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

$x = 5 i, - 5 i$

Step 5

The final solution is all the values that make $(3 x + 5) (x^{2} + 25) = 0$ true.

$x = - \frac{5}{3}, 5 i, - 5 i$

Step 6

Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^3+5x^2+75x+125

Related Questions

Support LCMGCF.com by disabling your adblocker.