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

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

Find the x-intercepts.

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

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

Solve the equation.

Set the numerator equal to zero.

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

Solve the equation for $x$ .

Factor the left side of the equation.

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

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

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

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

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

Factor $x$ out of $3 x$ .

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

Factor $x$ out of $x \cdot x^{2} + x (- 4 x)$ .

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

Factor $x$ out of $x (x^{2} - 4 x) + x \cdot 3$ .

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

Factor.

Factor $x^{2} - 4 x + 3$ 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 $3$ and whose sum is $- 4$ .

$- 3, - 1$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

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

$x = 0$

$x - 3 = 0$

$x - 1 = 0$

Set $x$ equal to $0$ .

$x = 0$

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 $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

$x = 0, 3, 1$

Exclude the solutions that do not make $0 = \frac{x^{3} - 4 x^{2} + 3 x}{x^{3} - x}$ true.

$x = 3$

x-intercept(s) in point form.

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

Find the y-intercepts.

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

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

The equation has an undefined fraction.

Undefined

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

y-intercept(s): $None$

List the intersections.

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

y-intercept(s): $None$

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

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