Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^2-x-4

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

Step 1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2, \pm 4$

$q = \pm 1, \pm 3$

Step 2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$3 {(- 1)}^{2} - (- 1) - 4$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 1$ is a root of the polynomial.

Simplify each term.

Raise $- 1$ to the power of $2$ .

$3 \cdot 1 - (- 1) - 4$

Multiply $3$ by $1$ .

$3 - (- 1) - 4$

Multiply $- 1$ by $- 1$ .

$3 + 1 - 4$

Simplify by adding and subtracting.

Add $3$ and $1$ .

$4 - 4$

Subtract $4$ from $4$ .

$0$

Step 5

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

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

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 1$	$3$	$- 1$	$- 4$
	\xa0		\xa0

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$3$	$- 1$	$- 4$
	\xa0		\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(- 1)$ and place the result of $(- 3)$ under the next term in the dividend $(- 1)$ .

$- 1$	$3$	$- 1$	$- 4$
	\xa0	$- 3$	\xa0
	$3$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$3$	$- 1$	$- 4$
	\xa0	$- 3$	\xa0
	$3$	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(- 1)$ and place the result of $(4)$ under the next term in the dividend $(- 4)$ .

$- 1$	$3$	$- 1$	$- 4$
	\xa0	$- 3$	$4$
	$3$	$- 4$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$3$	$- 1$	$- 4$
	\xa0	$- 3$	$4$
	$3$	$- 4$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$(3) x - 4$

Simplify the quotient polynomial.

$3 x - 4$

Step 7

Solve the equation to find any remaining roots.

Add $4$ to 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 $1$ .

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

Step 8

The polynomial can be written as a set of linear factors.

$(x + 1) (x - \frac{4}{3})$

Step 9

These are the roots (zeros) of the polynomial $3 x^{2} - x - 4$ .

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

Step 10

Find the Roots/Zeros Using the Rational Roots Test f(x)=3x^2-x-4

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