Find the Roots/Zeros Using the Rational Roots Test x^4-2x^3-240x-35

$x^{4} - 2 x^{3} - 240 x - 35$

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.

$p = \pm 1, \pm 5, \pm 7, \pm 35$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 5, \pm 7, \pm 35$

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.

${(7)}^{4} - 2 {(7)}^{3} - 240 \cdot 7 - 35$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 7$ is a root of the polynomial.

Simplify each term.

Raise $7$ to the power of $4$ .

$2401 - 2 {(7)}^{3} - 240 \cdot 7 - 35$

Raise $7$ to the power of $3$ .

$2401 - 2 \cdot 343 - 240 \cdot 7 - 35$

Multiply $- 2$ by $343$ .

$2401 - 686 - 240 \cdot 7 - 35$

Multiply $- 240$ by $7$ .

$2401 - 686 - 1680 - 35$

Simplify by subtracting numbers.

Subtract $686$ from $2401$ .

$1715 - 1680 - 35$

Subtract $1680$ from $1715$ .

$35 - 35$

Subtract $35$ from $35$ .

$0$

Step 5

Since $7$ is a known root, divide the polynomial by $x - 7$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 2 x^{3} - 240 x - 35}{x - 7}$

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.

$7$	$1$	$- 2$	$0$	$- 240$	$- 35$
	\xa0				\xa0

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

$7$	$1$	$- 2$	$0$	$- 240$	$- 35$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(7)$ and place the result of $(7)$ under the next term in the dividend $(- 2)$ .

$7$	$1$	$- 2$	$0$	$- 240$	$- 35$
	\xa0	$7$			\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.

$7$	$1$	$- 2$	$0$	$- 240$	$- 35$
	\xa0	$7$			\xa0
	$1$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(7)$ and place the result of $(35)$ under the next term in the dividend $(0)$ .

$7$	$1$	$- 2$	$0$	$- 240$	$- 35$
	\xa0	$7$	$35$		\xa0
	$1$	$5$

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

$7$	$1$	$- 2$	$0$	$- 240$	$- 35$
	\xa0	$7$	$35$		\xa0
	$1$	$5$	$35$

Multiply the newest entry in the result $(35)$ by the divisor $(7)$ and place the result of $(245)$ under the next term in the dividend $(- 240)$ .

$7$	$1$	$- 2$	$0$	$- 240$	$- 35$
	\xa0	$7$	$35$	$245$	\xa0
	$1$	$5$	$35$

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

$7$	$1$	$- 2$	$0$	$- 240$	$- 35$
	\xa0	$7$	$35$	$245$	\xa0
	$1$	$5$	$35$	$5$

Multiply the newest entry in the result $(5)$ by the divisor $(7)$ and place the result of $(35)$ under the next term in the dividend $(- 35)$ .

$7$	$1$	$- 2$	$0$	$- 240$	$- 35$
	\xa0	$7$	$35$	$245$	$35$
	$1$	$5$	$35$	$5$

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

$7$	$1$	$- 2$	$0$	$- 240$	$- 35$
	\xa0	$7$	$35$	$245$	$35$
	$1$	$5$	$35$	$5$	$0$

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

$1 x^{3} + 5 x^{2} + (35) x + 5$

Simplify the quotient polynomial.

$x^{3} + 5 x^{2} + 35 x + 5$

Step 7

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.14580561$

Step 8

The polynomial can be written as a set of linear factors.

$(x - 7) (x + 0.14580561)$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 2 x^{3} - 240 x - 35$ .

$x = 7, - 0.14580561$

Step 10

Find the Roots/Zeros Using the Rational Roots Test x^4-2x^3-240x-35

Related Questions

Support LCMGCF.com by disabling your adblocker.