Find the Roots (Zeros) 3x^3+28x^2+65x=0

$3 x^{3} + 28 x^{2} + 65 x = 0$

Step 1

Factor the left side of the equation.

Factor $x$ out of $3 x^{3} + 28 x^{2} + 65 x$ .

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

$x (3 x^{2}) + 28 x^{2} + 65 x = 0$

Factor $x$ out of $28 x^{2}$ .

$x (3 x^{2}) + x (28 x) + 65 x = 0$

Factor $x$ out of $65 x$ .

$x (3 x^{2}) + x (28 x) + x \cdot 65 = 0$

Factor $x$ out of $x (3 x^{2}) + x (28 x)$ .

$x (3 x^{2} + 28 x) + x \cdot 65 = 0$

Factor $x$ out of $x (3 x^{2} + 28 x) + x \cdot 65$ .

$x (3 x^{2} + 28 x + 65) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 65 = 195$ and whose sum is $b = 28$ .

Factor $28$ out of $28 x$ .

$x (3 x^{2} + 28 (x) + 65) = 0$

Rewrite $28$ as $13$ plus $15$

$x (3 x^{2} + (13 + 15) x + 65) = 0$

Apply the distributive property.

$x (3 x^{2} + 13 x + 15 x + 65) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x ((3 x^{2} + 13 x) + 15 x + 65) = 0$

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

$x (x (3 x + 13) + 5 (3 x + 13)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 13$ .

$x ((3 x + 13) (x + 5)) = 0$

Remove unnecessary parentheses.

$x (3 x + 13) (x + 5) = 0$

Step 2

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

$x = 0$

$3 x + 13 = 0$

$x + 5 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $3 x + 13$ equal to $0$ and solve for $x$ .

Set $3 x + 13$ equal to $0$ .

$3 x + 13 = 0$

Solve $3 x + 13 = 0$ for $x$ .

Subtract $13$ from both sides of the equation.

$3 x = - 13$

Divide each term in $3 x = - 13$ by $3$ and simplify.

Divide each term in $3 x = - 13$ by $3$ .

$\frac{3 x}{3} = \frac{- 13}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 13}{3}$

Divide $x$ by $1$ .

$x = \frac{- 13}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{13}{3}$

Step 5

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

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

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 6

The final solution is all the values that make $x (3 x + 13) (x + 5) = 0$ true.

$x = 0, - \frac{13}{3}, - 5$

Step 7

Find the Roots (Zeros) 3x^3+28x^2+65x=0

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