Chapter 4: Algebra and Functions

Indices, surds, and logarithms

Laws of indices: , , , .

Logarithms are the inverse of exponentials: (with , , ). Their laws are

and the change of base rule . Natural logarithms use base .

Surds are irrational roots. Rationalize denominators: .

Quadratics

For the roots are . The discriminant tells the nature of the roots: two distinct real roots; one repeated root; no real roots. If the roots are :

Any symmetric expression in the roots can be found from these; for example .

Worked example. For , and . Then . Factorizing, , so the roots are and ; check: ✓.

Inequalities and modulus

To solve for a rational or quadratic expression, find the critical values (zeros and undefined points), mark them on a number line, and test the sign in each interval. Never multiply or divide an inequality by an expression of unknown sign. For modulus, and or (for ).

Polynomials

The remainder theorem: when is divided by the remainder is . The factor theorem: is a factor of if and only if . A cubic with one known root can be reduced to a quadratic by division.

Partial fractions rewrite a rational function as a sum of simpler fractions, which is essential for integration. Example: write . Then . Put : . Put : , so . Hence .

Sequences and series

Standard sums: , , . Sums of this kind can often be found by telescoping after partial fractions.

The binomial theorem

The general term is . For and any real , .

Mathematical induction

To prove a statement for all positive integers: (1) base case: show ; (2) inductive step: assume and prove . Example: . Base: . Step: if the sum to terms is , then adding the next term gives ✓.

Functions

A function assigns to each element of the domain exactly one element of the range. Key ideas: composite ; inverse exists when is one-to-one and is the reflection of the graph in ; transformations of graphs (, , , , , ); even () and odd () functions; and the shapes of , , , and rational functions with their asymptotes.

Common mistakes

Dividing both sides of an inequality by a negative number without reversing the sign.

Forgetting the domain: needs .

Using when .

Writing .

Practice questions

Sum of the first 10 terms of the arithmetic series with , . [210]

Sum to infinity of . [9]

Coefficient of in . [80]

Prove by induction that .