Find the Roots/Zeros Using the Rational Roots Test f(x)=15x^3+113x^2-203x-45

$f (x) = 15 x^{3} + 113 x^{2} - 203 x - 45$

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.

$p = \pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45$

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

Step 2

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

$\pm 1, \pm \frac{1}{3}, \pm \frac{1}{5}, \pm \frac{1}{15}, \pm 3, \pm \frac{3}{5}, \pm 5, \pm \frac{5}{3}, \pm 9, \pm \frac{9}{5}, \pm 15, \pm 45$

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.

$15 {(- 9)}^{3} + 113 {(- 9)}^{2} - 203 \cdot - 9 - 45$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 9$ is a root of the polynomial.

Simplify each term.

Raise $- 9$ to the power of $3$ .

$15 \cdot - 729 + 113 {(- 9)}^{2} - 203 \cdot - 9 - 45$

Multiply $15$ by $- 729$ .

$- 10935 + 113 {(- 9)}^{2} - 203 \cdot - 9 - 45$

Raise $- 9$ to the power of $2$ .

$- 10935 + 113 \cdot 81 - 203 \cdot - 9 - 45$

Multiply $113$ by $81$ .

$- 10935 + 9153 - 203 \cdot - 9 - 45$

Multiply $- 203$ by $- 9$ .

$- 10935 + 9153 + 1827 - 45$

Simplify by adding and subtracting.

Add $- 10935$ and $9153$ .

$- 1782 + 1827 - 45$

Add $- 1782$ and $1827$ .

$45 - 45$

Subtract $45$ from $45$ .

$0$

Step 5

Since $- 9$ is a known root, divide the polynomial by $x + 9$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{15 x^{3} + 113 x^{2} - 203 x - 45}{x + 9}$

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.

$- 9$	$15$	$113$	$- 203$	$- 45$
	\xa0			\xa0

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

$- 9$	$15$	$113$	$- 203$	$- 45$
	\xa0			\xa0
	$15$

Multiply the newest entry in the result $(15)$ by the divisor $(- 9)$ and place the result of $(- 135)$ under the next term in the dividend $(113)$ .

$- 9$	$15$	$113$	$- 203$	$- 45$
	\xa0	$- 135$		\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.

$- 9$	$15$	$113$	$- 203$	$- 45$
	\xa0	$- 135$		\xa0
	$15$	$- 22$

Multiply the newest entry in the result $(- 22)$ by the divisor $(- 9)$ and place the result of $(198)$ under the next term in the dividend $(- 203)$ .

$- 9$	$15$	$113$	$- 203$	$- 45$
	\xa0	$- 135$	$198$	\xa0
	$15$	$- 22$

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

$- 9$	$15$	$113$	$- 203$	$- 45$
	\xa0	$- 135$	$198$	\xa0
	$15$	$- 22$	$- 5$

Multiply the newest entry in the result $(- 5)$ by the divisor $(- 9)$ and place the result of $(45)$ under the next term in the dividend $(- 45)$ .

$- 9$	$15$	$113$	$- 203$	$- 45$
	\xa0	$- 135$	$198$	$45$
	$15$	$- 22$	$- 5$

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

$- 9$	$15$	$113$	$- 203$	$- 45$
	\xa0	$- 135$	$198$	$45$
	$15$	$- 22$	$- 5$	$0$

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

$15 x^{2} + - 22 x - 5$

Simplify the quotient polynomial.

$15 x^{2} - 22 x - 5$

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 $a \cdot c = 15 \cdot - 5 = - 75$ and whose sum is $b = - 22$ .

Factor $- 22$ out of $- 22 x$ .

$(x + 9) + 15 x^{2} - 22 x - 5$

Rewrite $- 22$ as $3$ plus $- 25$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$(x + 9) + 3 x (5 x + 1) - 5 (5 x + 1)$

Factor the polynomial by factoring out the greatest common factor, $5 x + 1$ .

$(x + 9) + (5 x + 1) (3 x - 5)$

$(x + 9) (5 x + 1) (3 x - 5)$

Step 8

Factor the left side of the equation.

Factor $15 x^{3} + 113 x^{2} - 203 x - 45$ 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.

$p = \pm 1, \pm 45, \pm 3, \pm 15, \pm 5, \pm 9 q = \pm 1, \pm 15, \pm 3, \pm 5$

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

$\pm 1, \pm 0.0\overline{6}, \pm 0.\overline{3}, \pm 0.2, \pm 45, \pm 3, \pm 15, \pm 9, \pm 0.6, \pm 5, \pm 1.\overline{6}, \pm 1.8$

Substitute $- 0.2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.2$ is a root of the polynomial.

Substitute $- 0.2$ into the polynomial.

$15 {(- 0.2)}^{3} + 113 {(- 0.2)}^{2} - 203 \cdot - 0.2 - 45$

Raise $- 0.2$ to the power of $3$ .

$15 \cdot - 0.008 + 113 {(- 0.2)}^{2} - 203 \cdot - 0.2 - 45$

Multiply $15$ by $- 0.008$ .

$- 0.12 + 113 {(- 0.2)}^{2} - 203 \cdot - 0.2 - 45$

Raise $- 0.2$ to the power of $2$ .

$- 0.12 + 113 \cdot 0.04 - 203 \cdot - 0.2 - 45$

Multiply $113$ by $0.04$ .

$- 0.12 + 4.52 - 203 \cdot - 0.2 - 45$

Add $- 0.12$ and $4.52$ .

$4.4 - 203 \cdot - 0.2 - 45$

Multiply $- 203$ by $- 0.2$ .

$4.4 + 40.6 - 45$

Add $4.4$ and $40.6$ .

$45 - 45$

Subtract $45$ from $45$ .

$0$

Since $- 0.2$ is a known root, divide the polynomial by $5 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{15 x^{3} + 113 x^{2} - 203 x - 45}{5 x + 1}$

Divide $15 x^{3} + 113 x^{2} - 203 x - 45$ by $5 x + 1$ .

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

$5 x$

$1$

$15 x^{3}$

$113 x^{2}$

$203 x$

$45$

Divide the highest order term in the dividend $15 x^{3}$ by the highest order term in divisor $5 x$ .

					$3 x^{2}$
	$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			+	$15 x^{3}$	+	$3 x^{2}$

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

				$3 x^{2}$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$

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

				$3 x^{2}$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$
					+	$110 x^{2}$

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

				$3 x^{2}$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$
					+	$110 x^{2}$	-	$203 x$

Divide the highest order term in the dividend $110 x^{2}$ by the highest order term in divisor $5 x$ .

				$3 x^{2}$	+	$22 x$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$
					+	$110 x^{2}$	-	$203 x$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$22 x$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$
					+	$110 x^{2}$	-	$203 x$
					+	$110 x^{2}$	+	$22 x$

The expression needs to be subtracted from the dividend, so change all the signs in $110 x^{2} + 22 x$

				$3 x^{2}$	+	$22 x$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$
					+	$110 x^{2}$	-	$203 x$
					-	$110 x^{2}$	-	$22 x$

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

				$3 x^{2}$	+	$22 x$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$
					+	$110 x^{2}$	-	$203 x$
					-	$110 x^{2}$	-	$22 x$
							-	$225 x$

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

				$3 x^{2}$	+	$22 x$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$
					+	$110 x^{2}$	-	$203 x$
					-	$110 x^{2}$	-	$22 x$
							-	$225 x$	-	$45$

Divide the highest order term in the dividend $- 225 x$ by the highest order term in divisor $5 x$ .

				$3 x^{2}$	+	$22 x$	-	$45$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$
					+	$110 x^{2}$	-	$203 x$
					-	$110 x^{2}$	-	$22 x$
							-	$225 x$	-	$45$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$22 x$	-	$45$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$
					+	$110 x^{2}$	-	$203 x$
					-	$110 x^{2}$	-	$22 x$
							-	$225 x$	-	$45$
							-	$225 x$	-	$45$

The expression needs to be subtracted from the dividend, so change all the signs in $- 225 x - 45$

				$3 x^{2}$	+	$22 x$	-	$45$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$
					+	$110 x^{2}$	-	$203 x$
					-	$110 x^{2}$	-	$22 x$
							-	$225 x$	-	$45$
							+	$225 x$	+	$45$

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

				$3 x^{2}$	+	$22 x$	-	$45$
$5 x$	+	$1$		$15 x^{3}$	+	$113 x^{2}$	-	$203 x$	-	$45$
			-	$15 x^{3}$	-	$3 x^{2}$
					+	$110 x^{2}$	-	$203 x$
					-	$110 x^{2}$	-	$22 x$
							-	$225 x$	-	$45$
							+	$225 x$	+	$45$
										$0$

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

$3 x^{2} + 22 x - 45$

Write $15 x^{3} + 113 x^{2} - 203 x - 45$ as a set of factors.

$(5 x + 1) (3 x^{2} + 22 x - 45) = 0$

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 $a \cdot c = 3 \cdot - 45 = - 135$ and whose sum is $b = 22$ .

Factor $22$ out of $22 x$ .

$(5 x + 1) (3 x^{2} + 22 (x) - 45) = 0$

Rewrite $22$ as $- 5$ plus $27$

$(5 x + 1) (3 x^{2} + (- 5 + 27) x - 45) = 0$

Apply the distributive property.

$(5 x + 1) (3 x^{2} - 5 x + 27 x - 45) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(5 x + 1) ((3 x^{2} - 5 x) + 27 x - 45) = 0$

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

$(5 x + 1) (x (3 x - 5) + 9 (3 x - 5)) = 0$

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

$(5 x + 1) ((3 x - 5) (x + 9)) = 0$

Remove unnecessary parentheses.

$(5 x + 1) (3 x - 5) (x + 9) = 0$

Step 9

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

$5 x + 1 = 0$

$3 x - 5 = 0$

$x + 9 = 0$

Step 10

Set $5 x + 1$ equal to $0$ and solve for $x$ .

Set $5 x + 1$ equal to $0$ .

$5 x + 1 = 0$

Solve $5 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$5 x = - 1$

Divide each term in $5 x = - 1$ by $5$ and simplify.

Divide each term in $5 x = - 1$ by $5$ .

$\frac{5 x}{5} = \frac{- 1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 1}{5}$

Divide $x$ by $1$ .

$x = \frac{- 1}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{5}$

Step 11

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

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

$3 x - 5 = 0$

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

Add $5$ to both sides of the equation.

$3 x = 5$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 12

Set $x + 9$ equal to $0$ and solve for $x$ .

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 13

The final solution is all the values that make $(5 x + 1) (3 x - 5) (x + 9) = 0$ true.

$x = - \frac{1}{5}, \frac{5}{3}, - 9$

Step 14

Find the Roots/Zeros Using the Rational Roots Test f(x)=15x^3+113x^2-203x-45

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