Number and algebra
What this domain tests
Number and algebra underpins every other section of the 5-period paper. The exam expects fluency with real numbers, exact manipulation of surds and logarithms, and reliable solution of equations and inequalities; weak algebra costs marks in calculus and geometry questions too.
Core rules
- Number sets: N is contained in Z, which is contained in Q, which is contained in R. Numbers such as sqrt(2) and pi are irrational; keep exact forms unless a decimal is requested.
- Absolute value: |x| is the distance from 0. |a| = b with b >= 0 gives a = b or a = -b; |a| < b gives -b < a < b.
- Indices: a^m a^n = a^(m+n), (a^m)^n = a^(mn), a^(-n) = 1/a^n, a^(m/n) = the n-th root of a^m.
- Logarithms: log_a(xy) = log_a x + log_a y, log_a(x^n) = n log_a x, and log_a x requires x > 0.
- Polynomials: the remainder on dividing P(x) by (x - a) is P(a); (x - a) is a factor exactly when P(a) = 0.
- Quadratic: the discriminant b^2 - 4ac gives 0, 1 or 2 real roots; the sum of roots is -b/a and the product is c/a.
Worked example
Solve log_2 x + log_2(x - 3) = 2. Combine the logarithms: log_2[x(x - 3)] = 2, so x(x - 3) = 2^2 = 4. This gives x^2 - 3x - 4 = 0, that is (x - 4)(x + 1) = 0, so x = 4 or x = -1. The domain needs x > 0 and x - 3 > 0, hence x > 3. Reject x = -1. The solution is x = 4, and checking gives log_2 4 + log_2 1 = 2 + 0 = 2.
Traps that cost marks
- Squaring both sides of a radical equation can create extraneous roots; substitute every candidate back.
- Multiplying an inequality by a negative number reverses its sign, and you may not multiply by an expression whose sign is unknown.
- log(x + y) is not log x + log y; the product law applies only to products.
- Do not lose the plus-or-minus when solving x^2 = k, and state the domain before solving.