Question 1

When solving $|f(x)| = |g(x)|$, is squaring both sides always the best method?

Accepted Answer

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.

Question 2

What's the difference between the Remainder Theorem and the Factor Theorem?

Accepted Answer

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.

Question 3

Why do we learn partial fractions? What are they used for?

Accepted Answer

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.

Question 4

For the binomial expansion, what if the power is a positive integer?

Accepted Answer

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).

Algebra

What you need to know

The P2 Algebra Toolkit