finding roots?

Question

you're given the following table of values:

x              -10         -4          8          15

f(x)            -4          0          -4          -9

g(x)            5           9          0            4

 

and then you are given this y=xf(x)g(x)/ 1+f(x)^2 +g(x)^2 
what are the roots?

the answer give was (0, -4, 8) 

how did they reach this answer?

Answer 1

the roots of the equation y = 0 are obtained by setting the NUMERATOR of the expression for y to zero (a rational algebraic expression is zero if and only if its numerator is zero, for numbers in the domain of the function).

 

when x=0, the numerator is zero because x is one of the factors;

when x=-4, the factor f(x) is zero, and when x=8, the factor g(x) is zero.

 

Justified.  P.S. the denominator of the y definition is never zero.