Write an expression in one variable for the denominator of a rational expression, if the restrictions on the vairables are X doesnt equal 2, -3
Rational functions are functions that are polynomials divided by other polynomials. Any restrictions on the rational function are given by the denominator being set to zero.
In this case, the only restrictions on the function are that x cannot be 2 or -3, which means that the real factors of the denominator must be
`(x+3)` and `(x-2)` . Each of these factors may be raised to any positive power a and b. In addition, the denominator can have factors that have no real zeros, such as `x^2+1` or any real constants.
If the denominator is denoted by `d(x)` , then a possible denominator that satisfies the given conditions is `d(x)=a(x+3)^b(x-2)^c` where a is any real non-zero constant, and b, c are positive constants.
Suppose the rational expression is
f(x) = p(x)/q(x)
so if the restriction on x be such that x =/= 2, -3, then one such example can be,
q(x) = (x-2)^a (x+3)^b, where a,b are real numbers >0, one can simply take a=b=1, that is
q(x) = (x-2)(x+3), so this is a probable expression for the denominator of a rational expression where x cannot be equal to 2 and - 3, because at these two points the denominator q(x) of rational function f(x) becomes 0 making f(x) infinite !