Find the x and y Intercepts y=2x^3+x^2-7x-6

$y = 2 x^{3} + x^{2} - 7 x - 6$

Find the x-intercepts.

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

$0 = 2 x^{3} + x^{2} - 7 x - 6$

Solve the equation.

Rewrite the equation as $2 x^{3} + x^{2} - 7 x - 6 = 0$ .

$2 x^{3} + x^{2} - 7 x - 6 = 0$

Factor the left side of the equation.

Factor $2 x^{3} + x^{2} - 7 x - 6$ 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.

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

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

$\pm 1, \pm 0.5, \pm 6, \pm 3, \pm 2, \pm 1.5$

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.

$2 {(- 1)}^{3} + {(- 1)}^{2} - 7 \cdot - 1 - 6$

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

$2 \cdot - 1 + {(- 1)}^{2} - 7 \cdot - 1 - 6$

Multiply $2$ by $- 1$ .

$- 2 + {(- 1)}^{2} - 7 \cdot - 1 - 6$

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

$- 2 + 1 - 7 \cdot - 1 - 6$

Add $- 2$ and $1$ .

$- 1 - 7 \cdot - 1 - 6$

Multiply $- 7$ by $- 1$ .

$- 1 + 7 - 6$

Add $- 1$ and $7$ .

$6 - 6$

Subtract $6$ from $6$ .

$0$

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.

$\frac{2 x^{3} + x^{2} - 7 x - 6}{x + 1}$

Divide $2 x^{3} + x^{2} - 7 x - 6$ 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 $0$ .

$x$

$1$

$2 x^{3}$

$x^{2}$

$7 x$

$6$

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

					$2 x^{2}$
	$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			+	$2 x^{3}$	+	$2 x^{2}$

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$x^{2}$

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$7 x$

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

				$2 x^{2}$	-	$x$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$7 x$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$x$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$7 x$
					-	$x^{2}$	-	$x$

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

				$2 x^{2}$	-	$x$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$

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

				$2 x^{2}$	-	$x$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$

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

				$2 x^{2}$	-	$x$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$	-	$6$

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

				$2 x^{2}$	-	$x$	-	$6$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$	-	$6$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$x$	-	$6$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$	-	$6$
							-	$6 x$	-	$6$

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

				$2 x^{2}$	-	$x$	-	$6$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$	-	$6$
							+	$6 x$	+	$6$

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

				$2 x^{2}$	-	$x$	-	$6$
$x$	+	$1$		$2 x^{3}$	+	$x^{2}$	-	$7 x$	-	$6$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$x^{2}$	-	$7 x$
					+	$x^{2}$	+	$x$
							-	$6 x$	-	$6$
							+	$6 x$	+	$6$
										$0$

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

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

Write $2 x^{3} + x^{2} - 7 x - 6$ as a set of factors.

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

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

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

Rewrite $- 1$ as $3$ plus $- 4$

$(x + 1) (2 x^{2} + (3 - 4) x - 6) = 0$

Apply the distributive property.

$(x + 1) (2 x^{2} + 3 x - 4 x - 6) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 1) ((2 x^{2} + 3 x) - 4 x - 6) = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

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

Remove unnecessary parentheses.

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

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

$x + 1 = 0$

$2 x + 3 = 0$

$x - 2 = 0$

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 $2 x + 3$ equal to $0$ and solve for $x$ .

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Solve $2 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Divide each term in $2 x = - 3$ by $2$ and simplify.

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

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

$x = - 1, - \frac{3}{2}, 2$

x-intercept(s) in point form.

x-intercept(s): $(- 1, 0), (- \frac{3}{2}, 0), (2, 0)$

Find the y-intercepts.

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

$y = 2 {(0)}^{3} + {(0)}^{2} - 7 \cdot 0 - 6$

Solve the equation.

Remove parentheses.

$y = 2 \cdot 0^{3} + {(0)}^{2} - 7 \cdot 0 - 6$

Remove parentheses.

$y = 2 \cdot 0^{3} + 0^{2} - 7 \cdot 0 - 6$

Remove parentheses.

$y = 2 {(0)}^{3} + {(0)}^{2} - 7 \cdot 0 - 6$

Simplify $2 {(0)}^{3} + {(0)}^{2} - 7 \cdot 0 - 6$ .

Simplify each term.

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

$y = 2 \cdot 0 + {(0)}^{2} - 7 \cdot 0 - 6$

Multiply $2$ by $0$ .

$y = 0 + {(0)}^{2} - 7 \cdot 0 - 6$

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

$y = 0 + 0 - 7 \cdot 0 - 6$

Multiply $- 7$ by $0$ .

$y = 0 + 0 + 0 - 6$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 + 0 - 6$

Add $0$ and $0$ .

$y = 0 - 6$

Subtract $6$ from $0$ .

$y = - 6$

y-intercept(s) in point form.

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

List the intersections.

x-intercept(s): $(- 1, 0), (- \frac{3}{2}, 0), (2, 0)$

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

Find the x and y Intercepts y=2x^3+x^2-7x-6

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