Find the x and y Intercepts x^3-18x^2+107x-210

$x^{3} - 18 x^{2} + 107 x - 210$

Step 1

Write $x^{3} - 18 x^{2} + 107 x - 210$ as an equation.

$y = x^{3} - 18 x^{2} + 107 x - 210$

Step 2

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = x^{3} - 18 x^{2} + 107 x - 210$

Solve the equation.

Rewrite the equation as $x^{3} - 18 x^{2} + 107 x - 210 = 0$ .

$x^{3} - 18 x^{2} + 107 x - 210 = 0$

Factor the left side of the equation.

Factor $x^{3} - 18 x^{2} + 107 x - 210$ 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.

$p = \pm 1, \pm 210, \pm 2, \pm 105, \pm 3, \pm 70, \pm 5, \pm 42, \pm 6, \pm 35, \pm 7, \pm 30, \pm 10, \pm 21, \pm 14, \pm 15 q = \pm 1$

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

$\pm 1, \pm 210, \pm 2, \pm 105, \pm 3, \pm 70, \pm 5, \pm 42, \pm 6, \pm 35, \pm 7, \pm 30, \pm 10, \pm 21, \pm 14, \pm 15$

Substitute $5$ and simplify the expression. In this case, the expression is equal to $0$ so $5$ is a root of the polynomial.

Substitute $5$ into the polynomial.

$5^{3} - 18 \cdot 5^{2} + 107 \cdot 5 - 210$

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

$125 - 18 \cdot 5^{2} + 107 \cdot 5 - 210$

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

$125 - 18 \cdot 25 + 107 \cdot 5 - 210$

Multiply $- 18$ by $25$ .

$125 - 450 + 107 \cdot 5 - 210$

Subtract $450$ from $125$ .

$- 325 + 107 \cdot 5 - 210$

Multiply $107$ by $5$ .

$- 325 + 535 - 210$

Add $- 325$ and $535$ .

$210 - 210$

Subtract $210$ from $210$ .

$0$

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.

$\frac{x^{3} - 18 x^{2} + 107 x - 210}{x - 5}$

Divide $x^{3} - 18 x^{2} + 107 x - 210$ by $x - 5$ .

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

$x$

$5$

$x^{3}$

$18 x^{2}$

$107 x$

$210$

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

					$x^{2}$
	$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			+	$x^{3}$	-	$5 x^{2}$

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

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$

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

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$
					-	$13 x^{2}$

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

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$
					-	$13 x^{2}$	+	$107 x$

Divide the highest order term in the dividend $- 13 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$13 x$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$
					-	$13 x^{2}$	+	$107 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$13 x$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$
					-	$13 x^{2}$	+	$107 x$
					-	$13 x^{2}$	+	$65 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 13 x^{2} + 65 x$

				$x^{2}$	-	$13 x$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$
					-	$13 x^{2}$	+	$107 x$
					+	$13 x^{2}$	-	$65 x$

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

				$x^{2}$	-	$13 x$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$
					-	$13 x^{2}$	+	$107 x$
					+	$13 x^{2}$	-	$65 x$
							+	$42 x$

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

				$x^{2}$	-	$13 x$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$
					-	$13 x^{2}$	+	$107 x$
					+	$13 x^{2}$	-	$65 x$
							+	$42 x$	-	$210$

Divide the highest order term in the dividend $42 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$13 x$	+	$42$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$
					-	$13 x^{2}$	+	$107 x$
					+	$13 x^{2}$	-	$65 x$
							+	$42 x$	-	$210$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$13 x$	+	$42$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$
					-	$13 x^{2}$	+	$107 x$
					+	$13 x^{2}$	-	$65 x$
							+	$42 x$	-	$210$
							+	$42 x$	-	$210$

The expression needs to be subtracted from the dividend, so change all the signs in $42 x - 210$

				$x^{2}$	-	$13 x$	+	$42$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$
					-	$13 x^{2}$	+	$107 x$
					+	$13 x^{2}$	-	$65 x$
							+	$42 x$	-	$210$
							-	$42 x$	+	$210$

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

				$x^{2}$	-	$13 x$	+	$42$
$x$	-	$5$		$x^{3}$	-	$18 x^{2}$	+	$107 x$	-	$210$
			-	$x^{3}$	+	$5 x^{2}$
					-	$13 x^{2}$	+	$107 x$
					+	$13 x^{2}$	-	$65 x$
							+	$42 x$	-	$210$
							-	$42 x$	+	$210$
										$0$

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

$x^{2} - 13 x + 42$

Write $x^{3} - 18 x^{2} + 107 x - 210$ as a set of factors.

$(x - 5) (x^{2} - 13 x + 42) = 0$

Factor $x^{2} - 13 x + 42$ 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 $42$ and whose sum is $- 13$ .

$- 7, - 6$

Write the factored form using these integers.

$(x - 5) ((x - 7) (x - 6)) = 0$

Remove unnecessary parentheses.

$(x - 5) (x - 7) (x - 6) = 0$

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

$x - 5 = 0$

$x - 7 = 0$

$x - 6 = 0$

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.

$x = 5$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

The final solution is all the values that make $(x - 5) (x - 7) (x - 6) = 0$ true.

$x = 5, 7, 6$

x-intercept(s) in point form.

x-intercept(s): $(5, 0), (7, 0), (6, 0)$

Step 3

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = {(0)}^{3} - 18 {(0)}^{2} + 107 (0) - 210$

Solve the equation.

Remove parentheses.

$y = 0^{3} - 18 {(0)}^{2} + 107 (0) - 210$

Remove parentheses.

$y = 0^{3} - 18 \cdot 0^{2} + 107 (0) - 210$

Remove parentheses.

$y = {(0)}^{3} - 18 {(0)}^{2} + 107 (0) - 210$

Simplify ${(0)}^{3} - 18 {(0)}^{2} + 107 (0) - 210$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$y = 0 - 18 {(0)}^{2} + 107 (0) - 210$

Raising $0$ to any positive power yields $0$ .

$y = 0 - 18 \cdot 0 + 107 (0) - 210$

Multiply $- 18$ by $0$ .

$y = 0 + 0 + 107 (0) - 210$

Multiply $107$ by $0$ .

$y = 0 + 0 + 0 - 210$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 + 0 - 210$

Add $0$ and $0$ .

$y = 0 - 210$

Subtract $210$ from $0$ .

$y = - 210$

y-intercept(s) in point form.

y-intercept(s): $(0, - 210)$

Step 4

List the intersections.

x-intercept(s): $(5, 0), (7, 0), (6, 0)$

y-intercept(s): $(0, - 210)$

Step 5

Find the x and y Intercepts x^3-18x^2+107x-210

Related Questions

Support LCMGCF.com by disabling your adblocker.