Find the Roots/Zeros Using the Rational Roots Test 5x^3+7x^2+245x+343

$5 x^{3} + 7 x^{2} + 245 x + 343$

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.

$(5 x^{3} + 7 x^{2}) + 245 x + 343 = 0$

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

$x^{2} (5 x + 7) + 49 (5 x + 7) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 x + 7$ .

$(5 x + 7) (x^{2} + 49) = 0$

Step 2

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

$5 x + 7 = 0$

$x^{2} + 49 = 0$

Step 3

Set $5 x + 7$ equal to $0$ and solve for $x$ .

Set $5 x + 7$ equal to $0$ .

$5 x + 7 = 0$

Solve $5 x + 7 = 0$ for $x$ .

Subtract $7$ from both sides of the equation.

$5 x = - 7$

Divide each term in $5 x = - 7$ by $5$ and simplify.

Divide each term in $5 x = - 7$ by $5$ .

$\frac{5 x}{5} = \frac{- 7}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 7}{5}$

Divide $x$ by $1$ .

$x = \frac{- 7}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{7}{5}$

Step 4

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

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

$x^{2} + 49 = 0$

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

Subtract $49$ from both sides of the equation.

$x^{2} = - 49$

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

$x = \pm \sqrt{- 49}$

Simplify $\pm \sqrt{- 49}$ .

Rewrite $- 49$ as $- 1 (49)$ .

$x = \pm \sqrt{- 1 (49)}$

Rewrite $\sqrt{- 1 (49)}$ as $\sqrt{- 1} \cdot \sqrt{49}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{49}$

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

$x = \pm i \cdot \sqrt{49}$

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

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

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

$x = \pm i \cdot 7$

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

$x = \pm 7 i$

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.

$x = 7 i$

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

$x = - 7 i$

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

$x = 7 i, - 7 i$

Step 5

The final solution is all the values that make $(5 x + 7) (x^{2} + 49) = 0$ true.

$x = - \frac{7}{5}, 7 i, - 7 i$

Step 6

Find the Roots/Zeros Using the Rational Roots Test 5x^3+7x^2+245x+343

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