Find the Roots/Zeros Using the Rational Roots Test 3x^3-2x^2-75x+50

$3 x^{3} - 2 x^{2} - 75 x + 50$

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.

$p = \pm 1, \pm 2, \pm 5, \pm 10, \pm 25, \pm 50$

$q = \pm 1, \pm 3$

Step 2

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

$\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 5, \pm \frac{5}{3}, \pm 10, \pm \frac{10}{3}, \pm 25, \pm \frac{25}{3}, \pm 50, \pm \frac{50}{3}$

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.

$3 {(5)}^{3} - 2 {(5)}^{2} - 75 \cdot 5 + 50$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 5$ is a root of the polynomial.

Simplify each term.

Raise $5$ to the power of $3$ .

$3 \cdot 125 - 2 {(5)}^{2} - 75 \cdot 5 + 50$

Multiply $3$ by $125$ .

$375 - 2 {(5)}^{2} - 75 \cdot 5 + 50$

Raise $5$ to the power of $2$ .

$375 - 2 \cdot 25 - 75 \cdot 5 + 50$

Multiply $- 2$ by $25$ .

$375 - 50 - 75 \cdot 5 + 50$

Multiply $- 75$ by $5$ .

$375 - 50 - 375 + 50$

Simplify by adding and subtracting.

Subtract $50$ from $375$ .

$325 - 375 + 50$

Subtract $375$ from $325$ .

$- 50 + 50$

Add $- 50$ and $50$ .

$0$

Step 5

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{3} - 2 x^{2} - 75 x + 50}{x - 5}$

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.

$5$	$3$	$- 2$	$- 75$	$50$
	\xa0			\xa0

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

$5$	$3$	$- 2$	$- 75$	$50$
	\xa0			\xa0
	$3$

Multiply the newest entry in the result $(3)$ by the divisor $(5)$ and place the result of $(15)$ under the next term in the dividend $(- 2)$ .

$5$	$3$	$- 2$	$- 75$	$50$
	\xa0	$15$		\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.

$5$	$3$	$- 2$	$- 75$	$50$
	\xa0	$15$		\xa0
	$3$	$13$

Multiply the newest entry in the result $(13)$ by the divisor $(5)$ and place the result of $(65)$ under the next term in the dividend $(- 75)$ .

$5$	$3$	$- 2$	$- 75$	$50$
	\xa0	$15$	$65$	\xa0
	$3$	$13$

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

$5$	$3$	$- 2$	$- 75$	$50$
	\xa0	$15$	$65$	\xa0
	$3$	$13$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(5)$ and place the result of $(- 50)$ under the next term in the dividend $(50)$ .

$5$	$3$	$- 2$	$- 75$	$50$
	\xa0	$15$	$65$	$- 50$
	$3$	$13$	$- 10$

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

$5$	$3$	$- 2$	$- 75$	$50$
	\xa0	$15$	$65$	$- 50$
	$3$	$13$	$- 10$	$0$

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

$3 x^{2} + 13 x - 10$

Step 7

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 $a \cdot c = 3 \cdot - 10 = - 30$ and whose sum is $b = 13$ .

Factor $13$ out of $13 x$ .

$(x - 5) + 3 x^{2} + 13 (x) - 10$

Rewrite $13$ as $- 2$ plus $15$

$(x - 5) + 3 x^{2} + (- 2 + 15) x - 10$

Apply the distributive property.

$(x - 5) + 3 x^{2} - 2 x + 15 x - 10$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x - 5) (3 x^{2} - 2 x) + 15 x - 10$

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

$(x - 5) + x (3 x - 2) + 5 (3 x - 2)$

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$(x - 5) + (3 x - 2) (x + 5)$

$(x - 5) (3 x - 2) (x + 5)$

Step 8

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.

$(3 x^{3} - 2 x^{2}) - 75 x + 50 = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

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

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

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

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 $b = 5$ .

$(3 x - 2) ((x + 5) (x - 5)) = 0$

Remove unnecessary parentheses.

$(3 x - 2) (x + 5) (x - 5) = 0$

Step 9

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

$3 x - 2 = 0$

$x + 5 = 0$

$x - 5 = 0$

Step 10

Set $3 x - 2$ equal to $0$ and solve for $x$ .

Set $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Solve $3 x - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$3 x = 2$

Divide each term in $3 x = 2$ by $3$ and simplify.

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 11

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 12

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.

$x = 5$

Step 13

The final solution is all the values that make $(3 x - 2) (x + 5) (x - 5) = 0$ true.

$x = \frac{2}{3}, - 5, 5$

Step 14

Find the Roots/Zeros Using the Rational Roots Test 3x^3-2x^2-75x+50

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