Algebra — common mistakes

When finding the constants (A, B, C...), substituting strategic values of $x$ is often much faster than equating coefficients. For a term $\frac{A}{x-a}$, substituting $x=a$ into the cross-multiplied equation will quickly isolate $A$. Also, always check if the fraction is improper (degree of numerator $\ge$ degree of denominator) before you start. If it is, you must perform polynomial division first.

A very common question asks for the expansion and the range of values for which it is valid. Do not forget to state the validity range, e.g., $|x| < \frac{1}{2}$. This is an easy mark to gain and an easy one to lose. Be careful with signs, especially when $n$ is negative.

Squaring is a very reliable method because it correctly removes the modulus signs. However, it can lead to a more complicated polynomial to solve. The alternative method, setting $f(x) = g(x)$ and $f(x) = -g(x)$, can be quicker but requires careful management of signs and brackets. Both methods are valid; choose the one you are most comfortable and accurate with.

The Factor Theorem is simply a special case of the Remainder Theorem. The Remainder Theorem tells you the remainder when you divide a polynomial $P(x)$ by $(x-a)$ is $P(a)$. The Factor Theorem states that if this remainder $P(a)$ happens to be 0, then $(x-a)$ must be a factor.

The primary application of partial fractions in A-Level Mathematics is in integration (which you will study in depth). It is much easier to integrate a sum of simple fractions (like $\frac{2}{x-1} + \frac{3}{x+2}$) than it is to integrate the single complex fraction they came from. We also use them in binomial expansions.

If the power $n$ is a positive integer, the expansion formula still works, but it will terminate. The term with $(n-n)$ in the numerator will become zero, so all subsequent terms are zero. This gives a finite series, the same as the one you learned in P1 using combinations (nCr).