Answer:
The function [tex]f(x) = \frac{x^{2}-4}{x^{3}-3\cdot x^{2}-10\cdot x}[/tex] has a zero: [tex]x = 2[/tex], and two poles: [tex]x = 0[/tex] and [tex]x = 5[/tex].
Step-by-step explanation:
Let be [tex]f(x) = \frac{x^{2}-4}{x^{3}-3\cdot x^{2}-10\cdot x}[/tex] 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) [tex]f(x) = \frac{x^{2}-4}{x^{3}-3\cdot x^{2}-10\cdot x}[/tex] Given
2) [tex]f(x) = \frac{(x + 2)\cdot (x-2)}{x\cdot (x^{2}-3\cdot x -10)}[/tex] [tex]a^{2} - b^{2} = (a + b)\cdot (a - b)[/tex]/Associative, commutative and distributive properties.
3) [tex]f(x) = \frac{(x+2)\cdot (x-2)}{x\cdot (x-5)\cdot (x+2)}[/tex] [tex]x^{2} -(a+b)\cdot x + a\cdot b = (x-a)\cdot (x-b)[/tex]
4) [tex]f(x) = \frac{x-2}{x\cdot (x-5)}[/tex] Commutative property/Existence of multiplicative inverse/Modulative property/Result
The function [tex]f(x) = \frac{x^{2}-4}{x^{3}-3\cdot x^{2}-10\cdot x}[/tex] has a zero: [tex]x = 2[/tex], and two poles: [tex]x = 0[/tex] and [tex]x = 5[/tex].
