Find the x and y Intercepts f(x)=x^4-x^3-21x^2+x+20

$f (x) = x^{4} - x^{3} - 21 x^{2} + x + 20$

Find the x-intercepts.

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

$0 = x^{4} - x^{3} - 21 x^{2} + x + 20$

Solve the equation.

Rewrite the equation as $x^{4} - x^{3} - 21 x^{2} + x + 20 = 0$ .

$x^{4} - x^{3} - 21 x^{2} + x + 20 = 0$

Factor the left side of the equation.

Factor $x^{3}$ out of $x^{4} - x^{3}$ .

Factor $x^{3}$ out of $x^{4}$ .

$x^{3} x - x^{3} - 21 x^{2} + x + 20 = 0$

Factor $x^{3}$ out of $- x^{3}$ .

$x^{3} x + x^{3} \cdot - 1 - 21 x^{2} + x + 20 = 0$

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot - 1$ .

$x^{3} (x - 1) - 21 x^{2} + x + 20 = 0$

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 $a \cdot c = - 21 \cdot 20 = - 420$ and whose sum is $b = 1$ .

Multiply by $1$ .

$x^{3} (x - 1) - 21 x^{2} + 1 x + 20 = 0$

Rewrite $1$ as $- 20$ plus $21$

$x^{3} (x - 1) - 21 x^{2} + (- 20 + 21) x + 20 = 0$

Apply the distributive property.

$x^{3} (x - 1) - 21 x^{2} - 20 x + 21 x + 20 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x^{3} (x - 1) - 21 x^{2} - 20 x + 21 x + 20 = 0$

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

$x^{3} (x - 1) + x (- 21 x - 20) - (- 21 x - 20) = 0$

Factor the polynomial by factoring out the greatest common factor, $- 21 x - 20$ .

$x^{3} (x - 1) + (- 21 x - 20) (x - 1) = 0$

Factor $x - 1$ out of $x^{3} (x - 1) + (- 21 x - 20) (x - 1)$ .

Factor $x - 1$ out of $x^{3} (x - 1)$ .

$(x - 1) x^{3} + (- 21 x - 20) (x - 1) = 0$

Factor $x - 1$ out of $(- 21 x - 20) (x - 1)$ .

$(x - 1) x^{3} + (x - 1) (- 21 x - 20) = 0$

Factor $x - 1$ out of $(x - 1) x^{3} + (x - 1) (- 21 x - 20)$ .

$(x - 1) (x^{3} - 21 x - 20) = 0$

Factor.

Rewrite $x^{3} - 21 x - 20$ in a factored form.

Factor $x^{3} - 21 x - 20$ 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.

$p = \pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5 q = \pm 1$

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

$\pm 1, \pm 20, \pm 2, \pm 10, \pm 4, \pm 5$

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.

${(- 1)}^{3} - 21 \cdot - 1 - 20$

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

$- 1 - 21 \cdot - 1 - 20$

Multiply $- 21$ by $- 1$ .

$- 1 + 21 - 20$

Add $- 1$ and $21$ .

$20 - 20$

Subtract $20$ from $20$ .

$0$

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.

$\frac{x^{3} - 21 x - 20}{x + 1}$

Divide $x^{3} - 21 x - 20$ 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 $0$ .

$x$

$1$

$x^{3}$

$0 x^{2}$

$21 x$

$20$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$21 x$

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$21 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$21 x$
					-	$x^{2}$	-	$x$

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$21 x$
					+	$x^{2}$	+	$x$

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$21 x$
					+	$x^{2}$	+	$x$
							-	$20 x$

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$21 x$
					+	$x^{2}$	+	$x$
							-	$20 x$	-	$20$

Divide the highest order term in the dividend $- 20 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$x$	-	$20$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$21 x$
					+	$x^{2}$	+	$x$
							-	$20 x$	-	$20$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$20$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$21 x$
					+	$x^{2}$	+	$x$
							-	$20 x$	-	$20$
							-	$20 x$	-	$20$

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x - 20$

				$x^{2}$	-	$x$	-	$20$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$21 x$
					+	$x^{2}$	+	$x$
							-	$20 x$	-	$20$
							+	$20 x$	+	$20$

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

				$x^{2}$	-	$x$	-	$20$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	-	$21 x$	-	$20$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	-	$21 x$
					+	$x^{2}$	+	$x$
							-	$20 x$	-	$20$
							+	$20 x$	+	$20$
										$0$

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

$x^{2} - x - 20$

Write $x^{3} - 21 x - 20$ as a set of factors.

$(x - 1) ((x + 1) (x^{2} - x - 20)) = 0$

Factor $x^{2} - x - 20$ 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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Write the factored form using these integers.

$(x - 1) ((x + 1) ((x - 5) (x + 4))) = 0$

Remove unnecessary parentheses.

$(x - 1) ((x + 1) (x - 5) (x + 4)) = 0$

Remove unnecessary parentheses.

$(x - 1) (x + 1) (x - 5) (x + 4) = 0$

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

$x - 1 = 0$

$x + 1 = 0$

$x - 5 = 0$

$x + 4 = 0$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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.

$x = - 1$

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 $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(x - 1) (x + 1) (x - 5) (x + 4) = 0$ true.

$x = 1, - 1, 5, - 4$

x-intercept(s) in point form.

x-intercept(s): $(1, 0), (- 1, 0), (5, 0), (- 4, 0)$

Find the y-intercepts.

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

$y = {(0)}^{4} - {(0)}^{3} - 21 {(0)}^{2} + 0 + 20$

Solve the equation.

Remove parentheses.

$y = {(0)}^{4} - {(0)}^{3} - 21 {(0)}^{2} + 0 + 20$

Remove parentheses.

$y = 0^{4} - {(0)}^{3} - 21 {(0)}^{2} + 0 + 20$

Remove parentheses.

$y = 0^{4} - 0^{3} - 21 {(0)}^{2} + 0 + 20$

Remove parentheses.

$y = 0^{4} - 0^{3} - 21 \cdot 0^{2} + 0 + 20$

Remove parentheses.

$y = {(0)}^{4} - {(0)}^{3} - 21 {(0)}^{2} + 0 + 20$

Simplify ${(0)}^{4} - {(0)}^{3} - 21 {(0)}^{2} + 0 + 20$ .

Simplify each term.

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

$y = 0 - {(0)}^{3} - 21 {(0)}^{2} + 0 + 20$

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

$y = 0 - 0 - 21 {(0)}^{2} + 0 + 20$

Multiply $- 1$ by $0$ .

$y = 0 + 0 - 21 {(0)}^{2} + 0 + 20$

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

$y = 0 + 0 - 21 \cdot 0 + 0 + 20$

Multiply $- 21$ by $0$ .

$y = 0 + 0 + 0 + 0 + 20$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0 + 0 + 20$

Add $0$ and $0$ .

$y = 0 + 0 + 20$

Add $0$ and $0$ .

$y = 0 + 20$

Add $0$ and $20$ .

$y = 20$

y-intercept(s) in point form.

y-intercept(s): $(0, 20)$

List the intersections.

x-intercept(s): $(1, 0), (- 1, 0), (5, 0), (- 4, 0)$

y-intercept(s): $(0, 20)$

Find the x and y Intercepts f(x)=x^4-x^3-21x^2+x+20

Related Questions

Support LCMGCF.com by disabling your adblocker.