Advanced Algebra

This topic expands on foundational algebraic concepts, introducing tools essential for calculus and higher-level mathematics. It is divided into three core areas: the modulus function, manipulation of polynomials, and the decomposition of rational functions.

### 1. The Modulus Function

The modulus of a number, denoted by |x|, represents its absolute value or non-negative magnitude, irrespective of its sign. Formally, the modulus function is defined as:

* |x| = x, if x ≥ 0

* |x| = -x, if x < 0

For example, |5| = 5 and |-5| = 5. Graphically, the graph of y = |f(x)| is obtained by taking the graph of y = f(x) and reflecting any part that is below the x-axis in the x-axis itself. The graph of y = f(|x|) is obtained by keeping the part of the graph for x ≥ 0 and reflecting it in the y-axis, discarding the original part for x < 0.

Solving modulus equations and modulus inequalities is a key skill. For an equation like |ax + b| = c (where c ≥ 0), it implies two possibilities: ax + b = c or ax + b = -c. For an inequality like |ax + b| < c, this is equivalent to -c < ax + b < c. For |ax + b| > c, it means ax + b > c or ax + b < -c. An alternative powerful method, especially for more complex problems, is to square both sides of the equation or inequality, as this removes the modulus sign (since |x|² = x²). Care must be taken to ensure both sides are non-negative before squaring an inequality.

### 2. Polynomials: Division, Remainder, and Factor Theorems

A polynomial is an expression of the form P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0.

Polynomial division is a process similar to arithmetic long division, used to divide one polynomial by another of a lower degree. When a polynomial P(x) is divided by a divisor D(x), the result is a quotient Q(x) and a remainder R(x), related by the identity: P(x) ≡ D(x) * Q(x) + R(x). The degree of the remainder R(x) is always less than the degree of the divisor D(x).

Two critical theorems provide shortcuts for this process:

The Factor Theorem is incredibly useful for factorising cubic or higher-degree polynomials. The process is as follows:

* Use trial and error to find an integer root 'a' of P(x) = 0 by testing factors of the constant term.

* Once a root 'a' is found, you know (x - a) is a factor.

* Use polynomial long division to divide P(x) by (x - a) to obtain a quadratic quotient.

* Factorise the resulting quadratic quotient to find the remaining factors of P(x).

### 3. Partial Fractions

Partial fractions are used to decompose a complex rational function (a fraction with polynomials in the numerator and denominator) into a sum of simpler fractions. This is a vital technique for integration and other areas of advanced mathematics.

A key condition is that the rational function must be proper, meaning the degree of the numerator is less than the degree of the denominator. If it is improper, you must first perform polynomial division to express it as a polynomial plus a proper rational function.

The form of the partial fraction decomposition depends on the factors of the denominator:

* Case 1: Distinct Linear Factors: If the denominator is (ax + b)(cx + d), the decomposition is A/(ax + b) + B/(cx + d).

* Case 2: Repeated Linear Factors: If the denominator is (ax + b)², the decomposition is A/(ax + b) + B/(ax + b)².

* Case 3: Irreducible Quadratic Factor: If the denominator contains a factor like (ax² + bx + c) which cannot be factorised further, its corresponding part of the decomposition is (Ax + B)/(ax² + bx + c).

To find the unknown constants (A, B, etc.), you can use two main methods: