Core 4 - Partial Fractions

Question 1

As opposed to polynomial division, when are partial fractions useful?

Answer 1

When the largest power on the bottom is greater than the largest power on the top

Question 2

Outline the steps used to split 3/(x+4)(x+1) into partial fractions

Answer 2

EXPRESS AS PARTIAL FRACTIONS USING CONSTANTS A and B
3/(x+4)(x+1) = A/(x+4) + B/(x+1)

FIND COMMON DENOMINATOR
A(x+1)/(x+4)(x+1) + B(x+4)/(x+1)(x+4)

EQUATE TOP OF EACH SIDE
A(x+1) + B(x+4) = 3

EXPAND
Ax+A + Bx+4B = 3

2 METHODS:
-Compare Coefficients of x and the integer
A+B=0
A+4B=3
-Substitute in numbers to non expanded form to make A and B =0, e.g. x=-1, -4 to find values for A and B

Question 3

If there are three fractions and therefore three constants, how can you find the third value if no value of x cancels both other terms?

Answer 3

Find the values for the two constants, then sub in x=0, and the two values for the constants you know to find the final constant value.

Question 4

If there is a repeated root, what must you be careful of?

Answer 4

split up the partial fraction into ascending powers, e.g. (x+3), (x+3)^2.

Make sure the fraction’s common denominator is as simple as possible, so only multiply to get PQ^2 on the bottom, not PQQ^2, which you would get by cross multiplying the whole thing.

Question 5

What is the purpose of getting a partial fraction form to perform calculus?

Answer 5

It makes it easier to perform calculus, expecially integration, which often contains ln(x) because of the Ln trick.

Question 6

What form do you expect the partial fractions to be in if the co-efficient on top and bottom is the same?

Answer 6

A + B/(x-1) + C/(c+3)

There is always a singular constant A

Question 7

What is the quick method of finding the solution to a PF constant?

Answer 7

Use SHIFT + solve to solve the equation to reduce human error.

Question 8

When integrating from partial fractions, what must you remember with the powers?

Answer 8

Rewrite as negative power.