Hello there. To solve this question, we'll have to remember some properties about finding the common denominator in a sum of fractions.
Given the equation:
[tex]\frac{1}{x}+\frac{2}{x-3}=5[/tex]We have to determine its least common denominator.
For this, remember the definition for least common factor of polynomials:
If f(x) and g(x) are polynomials, their lcm(f(x), g(x)) can be calculated as:
[tex]lcm(f(x),g(x))=\dfrac{f(x)\cdot g(x)}{gcd(f(x),g(x))}[/tex]Where gcd(f(x),g(x)) is the greatest common divisor of the polynomials.
Usually, this expression gives another polynomial, as you can see we're dividing the product between f and g by their gcd.
In this case, notice f and g are linear functions. Most specifically they're first degree polynomials with leading coefficient equal 1 (monoic).
In this case, if f(x) is not equal to g(x), then we show that
[tex]gcd(f(x),g(x))=1[/tex]Such that their lcm is simply given by
[tex]lcm(f(x),g(x))=f(x)\cdot g(x)[/tex]Therefore we have the least common denominator of this equation as
[tex]lcm(x,x-3)=x\cdot(x-3)[/tex]This is the answer contained in the option b).
