Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3+4x^2-7x-10

$f (x) = x^{3} + 4 x^{2} - 7 x - 10$

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.

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

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 5, \pm 10$

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.

${(- 1)}^{3} + 4 {(- 1)}^{2} - 7 \cdot - 1 - 10$

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 $3$ .

$- 1 + 4 {(- 1)}^{2} - 7 \cdot - 1 - 10$

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

$- 1 + 4 \cdot 1 - 7 \cdot - 1 - 10$

Multiply $4$ by $1$ .

$- 1 + 4 - 7 \cdot - 1 - 10$

Multiply $- 7$ by $- 1$ .

$- 1 + 4 + 7 - 10$

Simplify by adding and subtracting.

Add $- 1$ and $4$ .

$3 + 7 - 10$

Add $3$ and $7$ .

$10 - 10$

Subtract $10$ from $10$ .

$0$

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.

$\frac{x^{3} + 4 x^{2} - 7 x - 10}{x + 1}$

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.

$- 1$	$1$	$4$	$- 7$	$- 10$
	\xa0			\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$1$	$4$	$- 7$	$- 10$
	\xa0			\xa0
	$1$

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

$- 1$	$1$	$4$	$- 7$	$- 10$
	\xa0	$- 1$		\xa0
	$1$

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

$- 1$	$1$	$4$	$- 7$	$- 10$
	\xa0	$- 1$		\xa0
	$1$	$3$

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 $(- 7)$ .

$- 1$	$1$	$4$	$- 7$	$- 10$
	\xa0	$- 1$	$- 3$	\xa0
	$1$	$3$

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

$- 1$	$1$	$4$	$- 7$	$- 10$
	\xa0	$- 1$	$- 3$	\xa0
	$1$	$3$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(- 1)$ and place the result of $(10)$ under the next term in the dividend $(- 10)$ .

$- 1$	$1$	$4$	$- 7$	$- 10$
	\xa0	$- 1$	$- 3$	$10$
	$1$	$3$	$- 10$

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

$- 1$	$1$	$4$	$- 7$	$- 10$
	\xa0	$- 1$	$- 3$	$10$
	$1$	$3$	$- 10$	$0$

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

$1 x^{2} + 3 x - 10$

Simplify the quotient polynomial.

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

Step 7

Factor $x^{2} + 3 x - 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$(x + 1) + (x - 2) (x + 5)$

$(x + 1) (x - 2) (x + 5)$

Step 8

Factor the left side of the equation.

Factor $x^{3} + 4 x^{2} - 7 x - 10$ 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.

$p = \pm 1, \pm 10, \pm 2, \pm 5 q = \pm 1$

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

$\pm 1, \pm 10, \pm 2, \pm 5$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} + 4 {(- 1)}^{2} - 7 \cdot - 1 - 10$

Raise $- 1$ to the power of $3$ .

$- 1 + 4 {(- 1)}^{2} - 7 \cdot - 1 - 10$

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

$- 1 + 4 \cdot 1 - 7 \cdot - 1 - 10$

Multiply $4$ by $1$ .

$- 1 + 4 - 7 \cdot - 1 - 10$

Add $- 1$ and $4$ .

$3 - 7 \cdot - 1 - 10$

Multiply $- 7$ by $- 1$ .

$3 + 7 - 10$

Add $3$ and $7$ .

$10 - 10$

Subtract $10$ from $10$ .

$0$

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.

$\frac{x^{3} + 4 x^{2} - 7 x - 10}{x + 1}$

Divide $x^{3} + 4 x^{2} - 7 x - 10$ by $x + 1$ .

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

$x$

$1$

$x^{3}$

$4 x^{2}$

$7 x$

$10$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$7 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$7 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$7 x$
					+	$3 x^{2}$	+	$3 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	-	$3 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	-	$3 x$
							-	$10 x$

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

				$x^{2}$	+	$3 x$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	-	$3 x$
							-	$10 x$	-	$10$

Divide the highest order term in the dividend $- 10 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$	-	$10$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	-	$3 x$
							-	$10 x$	-	$10$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	-	$10$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	-	$3 x$
							-	$10 x$	-	$10$
							-	$10 x$	-	$10$

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x - 10$

				$x^{2}$	+	$3 x$	-	$10$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	-	$3 x$
							-	$10 x$	-	$10$
							+	$10 x$	+	$10$

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

				$x^{2}$	+	$3 x$	-	$10$
$x$	+	$1$		$x^{3}$	+	$4 x^{2}$	-	$7 x$	-	$10$
			-	$x^{3}$	-	$x^{2}$
					+	$3 x^{2}$	-	$7 x$
					-	$3 x^{2}$	-	$3 x$
							-	$10 x$	-	$10$
							+	$10 x$	+	$10$
										$0$

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

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

Write $x^{3} + 4 x^{2} - 7 x - 10$ as a set of factors.

$(x + 1) (x^{2} + 3 x - 10) = 0$

Factor $x^{2} + 3 x - 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$(x + 1) ((x - 2) (x + 5)) = 0$

Remove unnecessary parentheses.

$(x + 1) (x - 2) (x + 5) = 0$

Step 9

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

$x + 1 = 0$

$x - 2 = 0$

$x + 5 = 0$

Step 10

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 11

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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 $(x + 1) (x - 2) (x + 5) = 0$ true.

$x = - 1, 2, - 5$

Step 14

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3+4x^2-7x-10

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