Find the x and y Intercepts f(x)=x^3-4x^2+x+6

$f (x) = x^{3} - 4 x^{2} + x + 6$

Find the x-intercepts.

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

$0 = x^{3} - 4 x^{2} + x + 6$

Solve the equation.

Rewrite the equation as $x^{3} - 4 x^{2} + x + 6 = 0$ .

$x^{3} - 4 x^{2} + x + 6 = 0$

Factor the left side of the equation.

Factor $x^{3} - 4 x^{2} + x + 6$ 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.

$p = \pm 1, \pm 6, \pm 2, \pm 3 q = \pm 1$

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

$\pm 1, \pm 6, \pm 2, \pm 3$

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.

${(- 1)}^{3} - 4 {(- 1)}^{2} - 1 + 6$

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

$- 1 - 4 {(- 1)}^{2} - 1 + 6$

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

$- 1 - 4 \cdot 1 - 1 + 6$

Multiply $- 4$ by $1$ .

$- 1 - 4 - 1 + 6$

Subtract $4$ from $- 1$ .

$- 5 - 1 + 6$

Subtract $1$ from $- 5$ .

$- 6 + 6$

Add $- 6$ and $6$ .

$0$

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.

$\frac{x^{3} - 4 x^{2} + x + 6}{x + 1}$

Divide $x^{3} - 4 x^{2} + x + 6$ 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 $0$ .

$x$

$1$

$x^{3}$

$4 x^{2}$

$x$

$6$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					-	$5 x^{2}$	-	$5 x$

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

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$

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

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$

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

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$	+	$6$

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$	+	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$	+	$6$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$	+	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$	+	$6$
							+	$6 x$	+	$6$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x + 6$

				$x^{2}$	-	$5 x$	+	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$	+	$6$
							-	$6 x$	-	$6$

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

				$x^{2}$	-	$5 x$	+	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$	+	$6$
							-	$6 x$	-	$6$
										$0$

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

$x^{2} - 5 x + 6$

Write $x^{3} - 4 x^{2} + x + 6$ as a set of factors.

$(x + 1) (x^{2} - 5 x + 6) = 0$

Factor $x^{2} - 5 x + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

$(x + 1) ((x - 3) (x - 2)) = 0$

Remove unnecessary parentheses.

$(x + 1) (x - 3) (x - 2) = 0$

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

$x + 1 = 0$

$x - 3 = 0$

$x - 2 = 0$

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 $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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.

$x = 2$

The final solution is all the values that make $(x + 1) (x - 3) (x - 2) = 0$ true.

$x = - 1, 3, 2$

x-intercept(s) in point form.

x-intercept(s): $(- 1, 0), (3, 0), (2, 0)$

Find the y-intercepts.

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

$y = {(0)}^{3} - 4 {(0)}^{2} + 0 + 6$

Solve the equation.

Remove parentheses.

$y = {(0)}^{3} - 4 {(0)}^{2} + 0 + 6$

Remove parentheses.

$y = 0^{3} - 4 {(0)}^{2} + 0 + 6$

Remove parentheses.

$y = 0^{3} - 4 \cdot 0^{2} + 0 + 6$

Remove parentheses.

$y = {(0)}^{3} - 4 {(0)}^{2} + 0 + 6$

Simplify ${(0)}^{3} - 4 {(0)}^{2} + 0 + 6$ .

Simplify each term.

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

$y = 0 - 4 {(0)}^{2} + 0 + 6$

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

$y = 0 - 4 \cdot 0 + 0 + 6$

Multiply $- 4$ by $0$ .

$y = 0 + 0 + 0 + 6$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0 + 6$

Add $0$ and $0$ .

$y = 0 + 6$

Add $0$ and $6$ .

$y = 6$

y-intercept(s) in point form.

y-intercept(s): $(0, 6)$

List the intersections.

x-intercept(s): $(- 1, 0), (3, 0), (2, 0)$

y-intercept(s): $(0, 6)$

Find the x and y Intercepts f(x)=x^3-4x^2+x+6

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