Find the Roots (Zeros) f(x)=(3x^2+x-4)/(x-1)

$f (x) = \frac{3 x^{2} + x - 4}{x - 1}$

Step 1

Set $\frac{3 x^{2} + x - 4}{x - 1}$ equal to $0$ .

$\frac{3 x^{2} + x - 4}{x - 1} = 0$

Step 2

Solve for $x$ .

Set the numerator equal to zero.

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

Solve the equation for $x$ .

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 $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 1$ .

Multiply by $1$ .

$3 x^{2} + 1 x - 4 = 0$

Rewrite $1$ as $- 3$ plus $4$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$3 x (x - 1) + 4 (x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$(x - 1) (3 x + 4) = 0$

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

$x - 1 = 0$

$3 x + 4 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $3 x + 4$ equal to $0$ and solve for $x$ .

Set $3 x + 4$ equal to $0$ .

$3 x + 4 = 0$

Solve $3 x + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Divide each term in $3 x = - 4$ by $3$ and simplify.

Divide each term in $3 x = - 4$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3}$

Divide $x$ by $1$ .

$x = \frac{- 4}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

The final solution is all the values that make $(x - 1) (3 x + 4) = 0$ true.

$x = 1, - \frac{4}{3}$

Exclude the solutions that do not make $\frac{3 x^{2} + x - 4}{x - 1} = 0$ true.

$x = - \frac{4}{3}$

Step 3

Find the Roots (Zeros) f(x)=(3x^2+x-4)/(x-1)

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