Find the Roots/Zeros Using the Rational Roots Test x^3+10x^2-25x-250

$x^{3} + 10 x^{2} - 25 x - 250$

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.

$p = \pm 1, \pm 2, \pm 5, \pm 10, \pm 25, \pm 50, \pm 125, \pm 250$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 5, \pm 10, \pm 25, \pm 50, \pm 125, \pm 250$

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.

${(5)}^{3} + 10 {(5)}^{2} - 25 \cdot 5 - 250$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 5$ is a root of the polynomial.

Simplify each term.

Raise $5$ to the power of $3$ .

$125 + 10 {(5)}^{2} - 25 \cdot 5 - 250$

Raise $5$ to the power of $2$ .

$125 + 10 \cdot 25 - 25 \cdot 5 - 250$

Multiply $10$ by $25$ .

$125 + 250 - 25 \cdot 5 - 250$

Multiply $- 25$ by $5$ .

$125 + 250 - 125 - 250$

Simplify by adding and subtracting.

Add $125$ and $250$ .

$375 - 125 - 250$

Subtract $125$ from $375$ .

$250 - 250$

Subtract $250$ from $250$ .

$0$

Step 5

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

$\frac{x^{3} + 10 x^{2} - 25 x - 250}{x - 5}$

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.

$5$	$1$	$10$	$- 25$	$- 250$
	\xa0			\xa0

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

$5$	$1$	$10$	$- 25$	$- 250$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(5)$ and place the result of $(5)$ under the next term in the dividend $(10)$ .

$5$	$1$	$10$	$- 25$	$- 250$
	\xa0	$5$		\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.

$5$	$1$	$10$	$- 25$	$- 250$
	\xa0	$5$		\xa0
	$1$	$15$

Multiply the newest entry in the result $(15)$ by the divisor $(5)$ and place the result of $(75)$ under the next term in the dividend $(- 25)$ .

$5$	$1$	$10$	$- 25$	$- 250$
	\xa0	$5$	$75$	\xa0
	$1$	$15$

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

$5$	$1$	$10$	$- 25$	$- 250$
	\xa0	$5$	$75$	\xa0
	$1$	$15$	$50$

Multiply the newest entry in the result $(50)$ by the divisor $(5)$ and place the result of $(250)$ under the next term in the dividend $(- 250)$ .

$5$	$1$	$10$	$- 25$	$- 250$
	\xa0	$5$	$75$	$250$
	$1$	$15$	$50$

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

$5$	$1$	$10$	$- 25$	$- 250$
	\xa0	$5$	$75$	$250$
	$1$	$15$	$50$	$0$

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

$1 x^{2} + 15 x + 50$

Simplify the quotient polynomial.

$x^{2} + 15 x + 50$

Step 7

Factor $x^{2} + 15 x + 50$ 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 $50$ and whose sum is $15$ .

$5, 10$

Write the factored form using these integers.

$(x - 5) + (x + 5) (x + 10)$

$(x - 5) (x + 5) (x + 10)$

Step 8

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{3} + 10 x^{2}) - 25 x - 250 = 0$

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

$x^{2} (x + 10) - 25 (x + 10) = 0$

Factor the polynomial by factoring out the greatest common factor, $x + 10$ .

$(x + 10) (x^{2} - 25) = 0$

Rewrite $25$ as $5^{2}$ .

$(x + 10) (x^{2} - 5^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$(x + 10) ((x + 5) (x - 5)) = 0$

Remove unnecessary parentheses.

$(x + 10) (x + 5) (x - 5) = 0$

Step 9

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

$x + 10 = 0$

$x + 5 = 0$

$x - 5 = 0$

Step 10

Set $x + 10$ equal to $0$ and solve for $x$ .

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 11

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 12

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

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 13

The final solution is all the values that make $(x + 10) (x + 5) (x - 5) = 0$ true.

$x = - 10, - 5, 5$

Step 14

Find the Roots/Zeros Using the Rational Roots Test x^3+10x^2-25x-250

Related Questions

Support LCMGCF.com by disabling your adblocker.