Algebra

General Binomial Expansion

If n is a positive integer there will be a finite number of terms (since
eventually there will be a factor of 0 in the numerator)j
This can be used for any values of n , although in all other cases this is an
infinite series
Can be used to find approximate values of x for a function by evaluating
the first few terms of the expansion (more terms = better approximation)
MUST state which values of x the expansion is valid for (ie (1 + a)n is valid
for
– 1 < a < 1

Partial Fraction

Fractions like this have two values that x cannot equal (ie if cx = – d then
the denominator would be 0, which cannot happen
To solve, substitute each of these values one at a time to remove either A or
B from the equation
Can also look at the coefficients on the denominator of the single fraction
to infer what A and B must be in order to get these coefficients

Partial Fractions – Repeated Root in Denominator

Fewer x values not allowed than unknown numerators cannot rely solely
on substituting alone
Therefore must also look at the coefficients on the numerator of the single
fraction to infer A, B and C
Same principle for roots to powers higher than 1 continue adding partial
fractions with the denominator power increasing by 1 until you reach the
power in the original fraction
Using Partial Fractions in Binomial Expansion
Can binomially expand single fraction with long denominator
Alternatively, could split fraction into partial fractions, binomially expand
each partial fraction and then sum each expansion
Second method often easier rather than multiplying different expansions
you need only sum them, which is more straightforward and less prone to
error