Find the x and y Intercepts r(x)=(x^2+3x-10)/((x+1)(x-3)(x+5))

$r (x) = \frac{x^{2} + 3 x - 10}{(x + 1) (x - 3) (x + 5)}$

Step 1

Find the x-intercepts.

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

$0 = \frac{x^{2} + 3 x - 10}{(x + 1) (x - 3) (x + 5)}$

Solve the equation.

Set the numerator equal to zero.

$x^{2} + 3 x - 10 = 0$

Solve the equation for $x$ .

Factor $x^{2} + 3 x - 10$ 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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

$(x - 2) (x + 5) = 0$

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

$x - 2 = 0$

$x + 5 = 0$

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.

$x = 2$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $(x - 2) (x + 5) = 0$ true.

$x = 2, - 5$

Exclude the solutions that do not make $0 = \frac{x^{2} + 3 x - 10}{(x + 1) (x - 3) (x + 5)}$ true.

$x = 2$

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

$y = \frac{{(0)}^{2} + 3 (0) - 10}{((0) + 1) ((0) - 3) ((0) + 5)}$

Solve the equation.

Remove parentheses.

$y = \frac{0^{2} + 3 (0) - 10}{((0) + 1) ((0) - 3) ((0) + 5)}$

Remove parentheses.

$y = \frac{0^{2} + 3 (0) - 10}{(0 + 1) ((0) - 3) ((0) + 5)}$

Remove parentheses.

$y = \frac{0^{2} + 3 (0) - 10}{(0 + 1) (0 - 3) ((0) + 5)}$

Remove parentheses.

$y = \frac{0^{2} + 3 (0) - 10}{(0 + 1) (0 - 3) (0 + 5)}$

Remove parentheses.

$y = \frac{{(0)}^{2} + 3 (0) - 10}{((0) + 1) ((0) - 3) ((0) + 5)}$

Simplify $\frac{{(0)}^{2} + 3 (0) - 10}{((0) + 1) ((0) - 3) ((0) + 5)}$ .

Simplify the numerator.

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

$y = \frac{0 + 3 (0) - 10}{(0 + 1) (0 - 3) (0 + 5)}$

Multiply $3$ by $0$ .

$y = \frac{0 + 0 - 10}{(0 + 1) (0 - 3) (0 + 5)}$

Add $0$ and $0$ .

$y = \frac{0 - 10}{(0 + 1) (0 - 3) (0 + 5)}$

Subtract $10$ from $0$ .

$y = \frac{- 10}{(0 + 1) (0 - 3) (0 + 5)}$

Simplify the denominator.

Add $0$ and $1$ .

$y = \frac{- 10}{1 (0 - 3) (0 + 5)}$

Multiply $0 - 3$ by $1$ .

$y = \frac{- 10}{(0 - 3) (0 + 5)}$

Subtract $3$ from $0$ .

$y = \frac{- 10}{- 3 (0 + 5)}$

Add $0$ and $5$ .

$y = \frac{- 10}{- 3 \cdot 5}$

Reduce the expression by cancelling the common factors.

Multiply $- 3$ by $5$ .

$y = \frac{- 10}{- 15}$

Cancel the common factor of $- 10$ and $- 15$ .

Factor $- 5$ out of $- 10$ .

$y = \frac{- 5 (2)}{- 15}$

Cancel the common factors.

Factor $- 5$ out of $- 15$ .

$y = \frac{- 5 \cdot 2}{- 5 \cdot 3}$

Cancel the common factor.

$y = \frac{- 5 \cdot 2}{- 5 \cdot 3}$

Rewrite the expression.

$y = \frac{2}{3}$

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4

Find the x and y Intercepts r(x)=(x^2+3x-10)/((x+1)(x-3)(x+5))

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