Revise Algebra for Cambridge 9709 (3.1)

The modulus of a number, denoted $|x|$, is its absolute value or magnitude. For example, $|-5| = 5$ and $|5| = 5$. In P3, you'll frequently solve equations and inequalities involving modulus functions. While algebraic cases can be used, a graphical approach is often more intuitive and less prone to error. Sketching the graphs of both sides of the equation or inequality allows you to visually identify the required intersection points or regions.

You will now learn to handle two more complex cases of partial fractions. The first is when the denominator has a repeated linear factor, such as $(x-2)^2$. The second is when the denominator contains an irreducible quadratic factor, meaning a quadratic that cannot be factorised into real linear factors, such as $x^2+9$. The key is to know the correct form to use for the decomposition.

Repeated Linear Factor: $\frac{f(x)}{(x-a)^2(x-b)} \equiv \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}$
Irreducible Quadratic Factor: $\frac{f(x)}{(x^2+c)(x-a)} \equiv \frac{Ax+B}{x^2+c} + \frac{C}{x-a}$

The binomial theorem can be extended to expand expressions of the form $(1+x)^n$ where $n$ is any rational number (positive, negative, or a fraction). Unlike the expansions from P1, these are infinite series. Consequently, they are only valid (i.e., they converge to the correct value) for a specific range of $x$ values.

For any rational number $n$, the expansion of $(1+x)^n$ is:
$1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots$
This expansion is valid for $|x| < 1$.

This section builds on the Factor and Remainder theorems. A key concept in P3 is that if a polynomial has real coefficients (which is always the case in A-Level questions unless stated otherwise), any complex roots must occur in conjugate pairs. This is a powerful tool for finding all roots of a polynomial when one complex root is known.