Question

Analyze the rational function: f(x) = (x^2 - 4)/(x^3 - 3x^2 - 10x)

Answer 1

The function
`f(x) = (x^(2)-4)/(x^(3)-3\cdot x^(2)-10\cdot x)` has a zero:
`x = 2`, and two poles:
`x = 0` and
`x = 5`.

Explanation:

Let be
`f(x) = (x^(2)-4)/(x^(3)-3\cdot x^(2)-10\cdot x)` a rational function armed with polynomial function. We proceed to factor the expression to the find the zeros and poles of the rational function:

1)
`f(x) = (x^(2)-4)/(x^(3)-3\cdot x^(2)-10\cdot x)` Given

2)
`f(x) = ((x + 2)\cdot (x-2))/(x\cdot (x^(2)-3\cdot x -10))`
`a^(2) - b^(2) = (a + b)\cdot (a - b)`/Associative, commutative and distributive properties.

3)
`f(x) = ((x+2)\cdot (x-2))/(x\cdot (x-5)\cdot (x+2))`
`x^(2) -(a+b)\cdot x + a\cdot b = (x-a)\cdot (x-b)`

4)
`f(x) = (x-2)/(x\cdot (x-5))` Commutative property/Existence of multiplicative inverse/Modulative property/Result

The function
`f(x) = (x^(2)-4)/(x^(3)-3\cdot x^(2)-10\cdot x)` has a zero:
`x = 2`, and two poles:
`x = 0` and
`x = 5`.