Numbers and Algebra
Numbers and Algebra tests fluency with number systems, algebraic manipulation, and equation solving. Questions combine rational and irrational numbers, exponents and logarithms, polynomials, inequalities, sequences, and matrices or complex numbers where the syllabus includes them.
Core rules
- For surds, simplify before combining: √a·√b=√(ab), and rationalise denominators by multiplying by the conjugate.
- Exponent laws require the same base; log_b(x) is defined only for x>0 and b>0, b≠1.
- Polynomial division and the factor theorem: if P(a)=0, then (x−a) is a factor.
- When multiplying or dividing an inequality by a negative number, reverse the sign.
- Arithmetic sequence: a_n=a_1+(n−1)d; geometric: a_n=a_1r^(n−1), with S_n=a_1(1−r^n)/(1−r), r≠1.
Worked example
Solve 2^(x+1)=8^(x−1). Write 8 as 2^3, so 8^(x−1)=2^(3x−3). Since bases are equal, x+1=3x−3. Then 4=2x, so x=2. Check: left 2^3=8; right 8^1=8.
Traps
Squaring both sides can introduce extraneous roots; always check in the original equation. Do not cancel a variable factor without considering that it may equal zero. For log equations, reject solutions outside the domain. In inequalities with absolute values, split into cases rather than dropping the bars. Sequence formulas use n terms, not n−1; check whether indexing starts at 0 or 1.