Represent in partial fraction form the expression x/x^2-3x+2

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Partial fractions are just another technique to simplify expressions to make them easier to manipulate in more complex questions. This is a low mark questions, testing your skills at using partial fractions - but dont worry! - once you master the technique all the questions are very similar.
To begin, all expressions to be simplified will either have an already factorised denominator, or factorisable, like the question we have above. If you find the denominator cant be factorised, then either there has been a mistake earlier in the question or you arn't intended to answer the question using partial fractions. Factorise the denominator
x/(x-1)(x-2)        (if the student struggles to factorise we would look at this for the rest of the session and move on to partial fractions once confident)
The next step is to write the equation in a generalised partial fraction form, the same form your answer is to be in, replacing the unknown numerators with constants to be found, like below
x/(x-1)(x-2) = A/(x-1) + B/(x-2)
From this point there are a few techniques to find the constants A and B. Are there any you have come across before in lessons which we could work from? (if yes, use their familiar method, if not use suggested method). The easiest way to find the constants is to equate the numerators of the two sides of the equation, as so, by multiplying through by both the denominators, giving
x = A(x-2) + B(x-1) 
From here we can substitute in values of x which will eliminate either the first A term or second B term in the expression, allowing us to find the constants A and B
substitute in x = 2
2 = (2-1)*B
B = 2
substitute in x = 1
1 = A*(1-2)
A = -1
We now have everything we need for the solution! Take the new found constants and go back to the generalised solution and substitute them in. The solution is
-1/(x-1) + 2/(x-2)
Previously research students exam board and see whether it is relevant to probe different twists on more complicated partial fractions once the simple process is mastered, eg 3 terms and constants, or numerators of the form Cx + D with higher powers in denominators.

Malachi R. GCSE Maths tutor, A Level Maths tutor, GCSE Further Mathem...

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