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

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

Find the x-intercepts.

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

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

Solve the equation.

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

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

Factor the left side of the equation.

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

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

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

Factor $x$ out of $6 x^{3}$ .

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

Factor $x$ out of $- x^{2}$ .

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

Factor $x$ out of $- 6 x$ .

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

Factor $x$ out of $x \cdot x^{3} + x (6 x^{2})$ .

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

Factor $x$ out of $x (x^{3} + 6 x^{2}) + x (- x)$ .

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

Factor $x$ out of $x (x^{3} + 6 x^{2} - x) + x \cdot - 6$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $x + 6$ .

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

Rewrite $1$ as $1^{2}$ .

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

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 $b = 1$ .

$x ((x + 6) ((x + 1) (x - 1))) = 0$

Remove unnecessary parentheses.

$x ((x + 6) (x + 1) (x - 1)) = 0$

Remove unnecessary parentheses.

$x (x + 6) (x + 1) (x - 1) = 0$

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

$x = 0$

$x + 6 = 0$

$x + 1 = 0$

$x - 1 = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $x (x + 6) (x + 1) (x - 1) = 0$ true.

$x = 0, - 6, - 1, 1$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 6, 0), (- 1, 0), (1, 0)$

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

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

Simplify each term.

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

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

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

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

Multiply $6$ by $0$ .

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

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

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

Multiply $- 1$ by $0$ .

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

Multiply $- 6$ by $0$ .

$y = 0 + 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0 + 0$

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

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

List the intersections.

x-intercept(s): $(0, 0), (- 6, 0), (- 1, 0), (1, 0)$

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

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

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