How can we derive equation 5.177 using the Remainder Theorem?

[htoo]

Could someone please explain me how can we get equation 5.177?
Thank you so much for your help

 

[Illmatic1]

!Equation 5.177 is an equation from a mathematics textbook. It states that if two polynomials, f(x) and g(x), are both divisible by the same factor (x-a) then their remainder, when divided by (x-a), is equal to the difference of the quotients of f(x) and g(x). Mathematically, this can be expressed as follows:(f(x) / (x-a)) - (g(x) / (x-a)) = f(a) - g(a) This equation is derived using the Remainder Theorem. According to the theorem, when a polynomial p(x) is divided by a linear factor (x-a), the remainder is equal to p(a). Therefore, in this case, the remainder for both f(x) and g(x) is equal to f(a) and g(a) respectively. Thus, we have:Remainder (f(x), (x-a)) = f(a) Remainder (g(x), (x-a)) = g(a) Subtracting these two equations yields the desired result.