Find the Roots/Zeros Using the Rational Roots Test f(x)=5x^3+7x^2-81x+45

$f (x) = 5 x^{3} + 7 x^{2} - 81 x + 45$

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.

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

$q = \pm 1, \pm 5$

Step 2

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

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

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.

$5 {(3)}^{3} + 7 {(3)}^{2} - 81 \cdot 3 + 45$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 3$ is a root of the polynomial.

Simplify each term.

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

$5 \cdot 27 + 7 {(3)}^{2} - 81 \cdot 3 + 45$

Multiply $5$ by $27$ .

$135 + 7 {(3)}^{2} - 81 \cdot 3 + 45$

Raise $3$ to the power of $2$ .

$135 + 7 \cdot 9 - 81 \cdot 3 + 45$

Multiply $7$ by $9$ .

$135 + 63 - 81 \cdot 3 + 45$

Multiply $- 81$ by $3$ .

$135 + 63 - 243 + 45$

Simplify by adding and subtracting.

Add $135$ and $63$ .

$198 - 243 + 45$

Subtract $243$ from $198$ .

$- 45 + 45$

Add $- 45$ and $45$ .

$0$

Step 5

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{5 x^{3} + 7 x^{2} - 81 x + 45}{x - 3}$

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.

$3$	$5$	$7$	$- 81$	$45$
	\xa0			\xa0

The first number in the dividend $(5)$ is put into the first position of the result area (below the horizontal line).

$3$	$5$	$7$	$- 81$	$45$
	\xa0			\xa0
	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(3)$ and place the result of $(15)$ under the next term in the dividend $(7)$ .

$3$	$5$	$7$	$- 81$	$45$
	\xa0	$15$		\xa0
	$5$

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

$3$	$5$	$7$	$- 81$	$45$
	\xa0	$15$		\xa0
	$5$	$22$

Multiply the newest entry in the result $(22)$ by the divisor $(3)$ and place the result of $(66)$ under the next term in the dividend $(- 81)$ .

$3$	$5$	$7$	$- 81$	$45$
	\xa0	$15$	$66$	\xa0
	$5$	$22$

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

$3$	$5$	$7$	$- 81$	$45$
	\xa0	$15$	$66$	\xa0
	$5$	$22$	$- 15$

Multiply the newest entry in the result $(- 15)$ by the divisor $(3)$ and place the result of $(- 45)$ under the next term in the dividend $(45)$ .

$3$	$5$	$7$	$- 81$	$45$
	\xa0	$15$	$66$	$- 45$
	$5$	$22$	$- 15$

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

$3$	$5$	$7$	$- 81$	$45$
	\xa0	$15$	$66$	$- 45$
	$5$	$22$	$- 15$	$0$

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

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

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

Factor $22$ out of $22 x$ .

$(x - 3) + 5 x^{2} + 22 (x) - 15$

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

Step 8

Factor the left side of the equation.

Factor $5 x^{3} + 7 x^{2} - 81 x + 45$ 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.

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

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

$\pm 1, \pm 0.2, \pm 45, \pm 9, \pm 3, \pm 0.6, \pm 15, \pm 5, \pm 1.8$

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

Substitute $3$ into the polynomial.

$5 \cdot 3^{3} + 7 \cdot 3^{2} - 81 \cdot 3 + 45$

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

$5 \cdot 27 + 7 \cdot 3^{2} - 81 \cdot 3 + 45$

Multiply $5$ by $27$ .

$135 + 7 \cdot 3^{2} - 81 \cdot 3 + 45$

Raise $3$ to the power of $2$ .

$135 + 7 \cdot 9 - 81 \cdot 3 + 45$

Multiply $7$ by $9$ .

$135 + 63 - 81 \cdot 3 + 45$

Add $135$ and $63$ .

$198 - 81 \cdot 3 + 45$

Multiply $- 81$ by $3$ .

$198 - 243 + 45$

Subtract $243$ from $198$ .

$- 45 + 45$

Add $- 45$ and $45$ .

$0$

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{5 x^{3} + 7 x^{2} - 81 x + 45}{x - 3}$

Divide $5 x^{3} + 7 x^{2} - 81 x + 45$ by $x - 3$ .

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

$x$

$3$

$5 x^{3}$

$7 x^{2}$

$81 x$

$45$

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

					$5 x^{2}$
	$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$

Multiply the new quotient term by the divisor.

				$5 x^{2}$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			+	$5 x^{3}$	-	$15 x^{2}$

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

				$5 x^{2}$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$

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

				$5 x^{2}$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$
					+	$22 x^{2}$

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

				$5 x^{2}$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$
					+	$22 x^{2}$	-	$81 x$

Divide the highest order term in the dividend $22 x^{2}$ by the highest order term in divisor $x$ .

				$5 x^{2}$	+	$22 x$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$
					+	$22 x^{2}$	-	$81 x$

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$22 x$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$
					+	$22 x^{2}$	-	$81 x$
					+	$22 x^{2}$	-	$66 x$

The expression needs to be subtracted from the dividend, so change all the signs in $22 x^{2} - 66 x$

				$5 x^{2}$	+	$22 x$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$
					+	$22 x^{2}$	-	$81 x$
					-	$22 x^{2}$	+	$66 x$

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

				$5 x^{2}$	+	$22 x$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$
					+	$22 x^{2}$	-	$81 x$
					-	$22 x^{2}$	+	$66 x$
							-	$15 x$

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

				$5 x^{2}$	+	$22 x$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$
					+	$22 x^{2}$	-	$81 x$
					-	$22 x^{2}$	+	$66 x$
							-	$15 x$	+	$45$

Divide the highest order term in the dividend $- 15 x$ by the highest order term in divisor $x$ .

				$5 x^{2}$	+	$22 x$	-	$15$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$
					+	$22 x^{2}$	-	$81 x$
					-	$22 x^{2}$	+	$66 x$
							-	$15 x$	+	$45$

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$22 x$	-	$15$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$
					+	$22 x^{2}$	-	$81 x$
					-	$22 x^{2}$	+	$66 x$
							-	$15 x$	+	$45$
							-	$15 x$	+	$45$

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x + 45$

				$5 x^{2}$	+	$22 x$	-	$15$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$
					+	$22 x^{2}$	-	$81 x$
					-	$22 x^{2}$	+	$66 x$
							-	$15 x$	+	$45$
							+	$15 x$	-	$45$

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

				$5 x^{2}$	+	$22 x$	-	$15$
$x$	-	$3$		$5 x^{3}$	+	$7 x^{2}$	-	$81 x$	+	$45$
			-	$5 x^{3}$	+	$15 x^{2}$
					+	$22 x^{2}$	-	$81 x$
					-	$22 x^{2}$	+	$66 x$
							-	$15 x$	+	$45$
							+	$15 x$	-	$45$
										$0$

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

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

Write $5 x^{3} + 7 x^{2} - 81 x + 45$ as a set of factors.

$(x - 3) (5 x^{2} + 22 x - 15) = 0$

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

Factor $22$ out of $22 x$ .

$(x - 3) (5 x^{2} + 22 (x) - 15) = 0$

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

$(x - 3) (5 x^{2} + (- 3 + 25) x - 15) = 0$

Apply the distributive property.

$(x - 3) (5 x^{2} - 3 x + 25 x - 15) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x - 3) ((5 x^{2} - 3 x) + 25 x - 15) = 0$

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

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

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

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

Remove unnecessary parentheses.

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

Step 9

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

$x - 3 = 0$

$5 x - 3 = 0$

$x + 5 = 0$

Step 10

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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 $x + 5$ 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 13

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

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

Step 14

Find the Roots/Zeros Using the Rational Roots Test f(x)=5x^3+7x^2-81x+45

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