Find the Roots (Zeros) f(x)=3x^3-7x^2-75x+175

$f (x) = 3 x^{3} - 7 x^{2} - 75 x + 175$

Step 1

Set $3 x^{3} - 7 x^{2} - 75 x + 175$ equal to $0$ .

$3 x^{3} - 7 x^{2} - 75 x + 175 = 0$

Step 2

Solve for $x$ .

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.

$(3 x^{3} - 7 x^{2}) - 75 x + 175 = 0$

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

$x^{2} (3 x - 7) - 25 (3 x - 7) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 7$ .

$(3 x - 7) (x^{2} - 25) = 0$

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

$(3 x - 7) (x^{2} - 5^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$(3 x - 7) ((x + 5) (x - 5)) = 0$

Remove unnecessary parentheses.

$(3 x - 7) (x + 5) (x - 5) = 0$

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

$3 x - 7 = 0$

$x + 5 = 0$

$x - 5 = 0$

Set $3 x - 7$ equal to $0$ and solve for $x$ .

Set $3 x - 7$ equal to $0$ .

$3 x - 7 = 0$

Solve $3 x - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$3 x = 7$

Divide each term in $3 x = 7$ by $3$ and simplify.

Divide each term in $3 x = 7$ by $3$ .

$\frac{3 x}{3} = \frac{7}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{7}{3}$

Divide $x$ by $1$ .

$x = \frac{7}{3}$

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.

$x = - 5$

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

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $(3 x - 7) (x + 5) (x - 5) = 0$ true.

$x = \frac{7}{3}, - 5, 5$

Step 3

Find the Roots (Zeros) f(x)=3x^3-7x^2-75x+175

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