Find the Roots/Zeros Using the Rational Roots Test f(x)=15x^3+371x^2-103x-75

$f (x) = 15 x^{3} + 371 x^{2} - 103 x - 75$

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.

$p = \pm 1, \pm 3, \pm 5, \pm 15, \pm 25, \pm 75$

$q = \pm 1, \pm 3, \pm 5, \pm 15$

Step 2

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

$\pm 1, \pm \frac{1}{3}, \pm \frac{1}{5}, \pm \frac{1}{15}, \pm 3, \pm \frac{3}{5}, \pm 5, \pm \frac{5}{3}, \pm 15, \pm 25, \pm \frac{25}{3}, \pm 75$

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.

$15 {(- 25)}^{3} + 371 {(- 25)}^{2} - 103 \cdot - 25 - 75$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 25$ is a root of the polynomial.

Simplify each term.

Raise $- 25$ to the power of $3$ .

$15 \cdot - 15625 + 371 {(- 25)}^{2} - 103 \cdot - 25 - 75$

Multiply $15$ by $- 15625$ .

$- 234375 + 371 {(- 25)}^{2} - 103 \cdot - 25 - 75$

Raise $- 25$ to the power of $2$ .

$- 234375 + 371 \cdot 625 - 103 \cdot - 25 - 75$

Multiply $371$ by $625$ .

$- 234375 + 231875 - 103 \cdot - 25 - 75$

Multiply $- 103$ by $- 25$ .

$- 234375 + 231875 + 2575 - 75$

Simplify by adding and subtracting.

Add $- 234375$ and $231875$ .

$- 2500 + 2575 - 75$

Add $- 2500$ and $2575$ .

$75 - 75$

Subtract $75$ from $75$ .

$0$

Step 5

Since $- 25$ is a known root, divide the polynomial by $x + 25$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{15 x^{3} + 371 x^{2} - 103 x - 75}{x + 25}$

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.

$- 25$	$15$	$371$	$- 103$	$- 75$
	\xa0			\xa0

The first number in the dividend $(15)$ is put into the first position of the result area (below the horizontal line).

$- 25$	$15$	$371$	$- 103$	$- 75$
	\xa0			\xa0
	$15$

Multiply the newest entry in the result $(15)$ by the divisor $(- 25)$ and place the result of $(- 375)$ under the next term in the dividend $(371)$ .

$- 25$	$15$	$371$	$- 103$	$- 75$
	\xa0	$- 375$		\xa0
	$15$

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

$- 25$	$15$	$371$	$- 103$	$- 75$
	\xa0	$- 375$		\xa0
	$15$	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(- 25)$ and place the result of $(100)$ under the next term in the dividend $(- 103)$ .

$- 25$	$15$	$371$	$- 103$	$- 75$
	\xa0	$- 375$	$100$	\xa0
	$15$	$- 4$

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

$- 25$	$15$	$371$	$- 103$	$- 75$
	\xa0	$- 375$	$100$	\xa0
	$15$	$- 4$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 25)$ and place the result of $(75)$ under the next term in the dividend $(- 75)$ .

$- 25$	$15$	$371$	$- 103$	$- 75$
	\xa0	$- 375$	$100$	$75$
	$15$	$- 4$	$- 3$

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

$- 25$	$15$	$371$	$- 103$	$- 75$
	\xa0	$- 375$	$100$	$75$
	$15$	$- 4$	$- 3$	$0$

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

$15 x^{2} + - 4 x - 3$

Simplify the quotient polynomial.

$15 x^{2} - 4 x - 3$

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 $a \cdot c = 15 \cdot - 3 = - 45$ and whose sum is $b = - 4$ .

Factor $- 4$ out of $- 4 x$ .

$(x + 25) + 15 x^{2} - 4 x - 3$

Rewrite $- 4$ as $5$ plus $- 9$

$(x + 25) + 15 x^{2} + (5 - 9) x - 3$

Apply the distributive property.

$(x + 25) + 15 x^{2} + 5 x - 9 x - 3$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 25) (15 x^{2} + 5 x) - 9 x - 3$

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

$(x + 25) + 5 x (3 x + 1) - 3 (3 x + 1)$

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$(x + 25) + (3 x + 1) (5 x - 3)$

$(x + 25) (3 x + 1) (5 x - 3)$

Step 8

Factor the left side of the equation.

Factor $15 x^{3} + 371 x^{2} - 103 x - 75$ using the rational roots test.

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.

$p = \pm 1, \pm 75, \pm 3, \pm 25, \pm 5, \pm 15 q = \pm 1, \pm 15, \pm 3, \pm 5$

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

$\pm 1, \pm 0.0\overline{6}, \pm 0.\overline{3}, \pm 0.2, \pm 75, \pm 5, \pm 25, \pm 15, \pm 3, \pm 0.6, \pm 1.\overline{6}, \pm 8.\overline{3}$

Substitute $- 0.\overline{3}$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.\overline{3}$ is a root of the polynomial.

Substitute $- 0.\overline{3}$ into the polynomial.

$15 {(- 0.\overline{3})}^{3} + 371 {(- 0.\overline{3})}^{2} - 103 \cdot - 0.\overline{3} - 75$

Raise $- 0.\overline{3}$ to the power of $3$ .

$15 \cdot - 0.\overline{037} + 371 {(- 0.\overline{3})}^{2} - 103 \cdot - 0.\overline{3} - 75$

Multiply $15$ by $- 0.\overline{037}$ .

$- 0.\overline{5} + 371 {(- 0.\overline{3})}^{2} - 103 \cdot - 0.\overline{3} - 75$

Raise $- 0.\overline{3}$ to the power of $2$ .

$- 0.\overline{5} + 371 \cdot 0.\overline{1} - 103 \cdot - 0.\overline{3} - 75$

Multiply $371$ by $0.\overline{1}$ .

$- 0.\overline{5} + 41.22222222 - 103 \cdot - 0.\overline{3} - 75$

Add $- 0.\overline{5}$ and $41.22222222$ .

$40.66666666 - 103 \cdot - 0.\overline{3} - 75$

Multiply $- 103$ by $- 0.\overline{3}$ .

$40.66666666 + 34.\overline{3} - 75$

Add $40.66666666$ and $34.\overline{3}$ .

$75 - 75$

Subtract $75$ from $75$ .

$0$

Since $- 0.\overline{3}$ is a known root, divide the polynomial by $3 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{15 x^{3} + 371 x^{2} - 103 x - 75}{3 x + 1}$

Divide $15 x^{3} + 371 x^{2} - 103 x - 75$ by $3 x + 1$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$3 x$

$1$

$15 x^{3}$

$371 x^{2}$

$103 x$

$75$

Divide the highest order term in the dividend $15 x^{3}$ by the highest order term in divisor $3 x$ .

					$5 x^{2}$
	$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$

Multiply the new quotient term by the divisor.

				$5 x^{2}$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			+	$15 x^{3}$	+	$5 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $15 x^{3} + 5 x^{2}$

				$5 x^{2}$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$5 x^{2}$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$
					+	$366 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$5 x^{2}$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$
					+	$366 x^{2}$	-	$103 x$

Divide the highest order term in the dividend $366 x^{2}$ by the highest order term in divisor $3 x$ .

				$5 x^{2}$	+	$122 x$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$
					+	$366 x^{2}$	-	$103 x$

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$122 x$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$
					+	$366 x^{2}$	-	$103 x$
					+	$366 x^{2}$	+	$122 x$

The expression needs to be subtracted from the dividend, so change all the signs in $366 x^{2} + 122 x$

				$5 x^{2}$	+	$122 x$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$
					+	$366 x^{2}$	-	$103 x$
					-	$366 x^{2}$	-	$122 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$5 x^{2}$	+	$122 x$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$
					+	$366 x^{2}$	-	$103 x$
					-	$366 x^{2}$	-	$122 x$
							-	$225 x$

Pull the next terms from the original dividend down into the current dividend.

				$5 x^{2}$	+	$122 x$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$
					+	$366 x^{2}$	-	$103 x$
					-	$366 x^{2}$	-	$122 x$
							-	$225 x$	-	$75$

Divide the highest order term in the dividend $- 225 x$ by the highest order term in divisor $3 x$ .

				$5 x^{2}$	+	$122 x$	-	$75$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$
					+	$366 x^{2}$	-	$103 x$
					-	$366 x^{2}$	-	$122 x$
							-	$225 x$	-	$75$

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$122 x$	-	$75$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$
					+	$366 x^{2}$	-	$103 x$
					-	$366 x^{2}$	-	$122 x$
							-	$225 x$	-	$75$
							-	$225 x$	-	$75$

The expression needs to be subtracted from the dividend, so change all the signs in $- 225 x - 75$

				$5 x^{2}$	+	$122 x$	-	$75$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$
					+	$366 x^{2}$	-	$103 x$
					-	$366 x^{2}$	-	$122 x$
							-	$225 x$	-	$75$
							+	$225 x$	+	$75$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$5 x^{2}$	+	$122 x$	-	$75$
$3 x$	+	$1$		$15 x^{3}$	+	$371 x^{2}$	-	$103 x$	-	$75$
			-	$15 x^{3}$	-	$5 x^{2}$
					+	$366 x^{2}$	-	$103 x$
					-	$366 x^{2}$	-	$122 x$
							-	$225 x$	-	$75$
							+	$225 x$	+	$75$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$5 x^{2} + 122 x - 75$

Write $15 x^{3} + 371 x^{2} - 103 x - 75$ as a set of factors.

$(3 x + 1) (5 x^{2} + 122 x - 75) = 0$

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 $a \cdot c = 5 \cdot - 75 = - 375$ and whose sum is $b = 122$ .

Factor $122$ out of $122 x$ .

$(3 x + 1) (5 x^{2} + 122 (x) - 75) = 0$

Rewrite $122$ as $- 3$ plus $125$

$(3 x + 1) (5 x^{2} + (- 3 + 125) x - 75) = 0$

Apply the distributive property.

$(3 x + 1) (5 x^{2} - 3 x + 125 x - 75) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x + 1) ((5 x^{2} - 3 x) + 125 x - 75) = 0$

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

$(3 x + 1) (x (5 x - 3) + 25 (5 x - 3)) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 x - 3$ .

$(3 x + 1) ((5 x - 3) (x + 25)) = 0$

Remove unnecessary parentheses.

$(3 x + 1) (5 x - 3) (x + 25) = 0$

Step 9

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

$3 x + 1 = 0$

$5 x - 3 = 0$

$x + 25 = 0$

Step 10

Set $3 x + 1$ equal to $0$ and solve for $x$ .

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Solve $3 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Divide each term in $3 x = - 1$ by $3$ and simplify.

Divide each term in $3 x = - 1$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 11

Set $5 x - 3$ equal to $0$ and solve for $x$ .

Set $5 x - 3$ equal to $0$ .

$5 x - 3 = 0$

Solve $5 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$5 x = 3$

Divide each term in $5 x = 3$ by $5$ and simplify.

Divide each term in $5 x = 3$ by $5$ .

$\frac{5 x}{5} = \frac{3}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{3}{5}$

Divide $x$ by $1$ .

$x = \frac{3}{5}$

Step 12

Set $x + 25$ equal to $0$ and solve for $x$ .

Set $x + 25$ equal to $0$ .

$x + 25 = 0$

Subtract $25$ from both sides of the equation.

$x = - 25$

Step 13

The final solution is all the values that make $(3 x + 1) (5 x - 3) (x + 25) = 0$ true.

$x = - \frac{1}{3}, \frac{3}{5}, - 25$

Step 14

Find the Roots/Zeros Using the Rational Roots Test f(x)=15x^3+371x^2-103x-75

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