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

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

Step 1

Find the x-intercepts.

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

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

Solve the equation.

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

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

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

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

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 $b = 1$ .

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

Remove unnecessary parentheses.

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

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

$x - 3 = 0$

$x + 1 = 0$

$x - 1 = 0$

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

$x = 3, - 1, 1$

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

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

Simplify each term.

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

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

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

$y = 0 - 3 \cdot 0 - (0) + 3$

Multiply $- 3$ by $0$ .

$y = 0 + 0 - (0) + 3$

Multiply $- 1$ by $0$ .

$y = 0 + 0 + 0 + 3$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0 + 3$

Add $0$ and $0$ .

$y = 0 + 3$

Add $0$ and $3$ .

$y = 3$

y-intercept(s) in point form.

y-intercept(s): $(0, 3)$

Step 3

List the intersections.

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

y-intercept(s): $(0, 3)$

Step 4

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

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