What is an improper fraction, and how to I make thisproper so that it can be differentiated?

An improper fraction is any fraction where the order of the constant (power of x) is equal or greater in the nuberator than the denominator, take for example;

(x2-3x-2)/(x2-3x+2)

This can first be factorised to give (x2-3x-2)/(x2-3x+2)=(x2-3x-2)/(x-1)(x-2).

We then set this fraction to be equal to a series of constants, divided by the factors i.e

(x2-3x-2)/(x-1)(x-2) = A/(x-1) + B/(x-2).  Multiplying by the denominator gives us;

(x2-3x-2) = A(x-2) + B(x-1).  This can be solved for A and B by substitution of  values.  The best place to start is by setting one factor equal to 0, in this case we first let x=2 so that the A term is 0, and find B=-4.  We then set x=1 so that the B term=0 to find A=4.  We can then write:

(x2-3x-2)/(x2-3x+2) = 4/(x-1) -4/(x-2).  Which is an easier form to handle for further calculation such as differentiation.

Answered by Aidan S. Maths tutor

3187 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Find the equation of the tangent to the curve y = 2 ln(2e - x) at the point on the curve where x = e.


The quadratic equation (k+1)x^2 + (5k-3)x + 3k = 0 has equal roots, find the possible values of the real number k.


Is there an easy way to remember all the basic graphical transformations?


Find the acute angle between the two lines... l1: r = (4, 28, 4) + λ(-1, -5, 1), l2: r = (5, 3, 1) + μ(3, 0, -4)


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences