Algebra
What the domain tests
MSCE Additional Mathematics algebra rewards fluent manipulation of indices, surds, logarithms, quadratic theory, polynomials and inequalities. Items often combine two or three skills, so a correct final answer needs visible lawful steps; a bare result can lose method marks even when the value is right.
Core rules
- For a>0, a^m a^n=a^{m+n}, (a^m)^n=a^{mn}, a^0=1 and a^{-n}=1/a^n. Apply these before substituting numerical values.
- log_a(xy)=log_a x+log_a y; log_a(x/y)=log_a x-log_a y; log_a(x^n)=n log_a x. Every logarithm requires a positive argument.
- For ax^2+bx+c=0, discriminant Δ=b^2-4ac. Roots are real and distinct if Δ>0, equal if Δ=0, and non-real if Δ<0. Sum of roots=-b/a; product=c/a.
- Factor theorem: (x-k) is a factor of P(x) iff P(k)=0; the remainder on division by x-k is P(k).
- Rationalise a denominator by multiplying numerator and denominator by the conjugate.
Worked example
Solve log_2(x+3)+log_2(x-1)=3. Combine the logs: log_2[(x+3)(x-1)]=3, so (x+3)(x-1)=2^3=8. Expanding gives x^2+2x-3=8, hence x^2+2x-11=0. The quadratic formula gives x=-1±2√3. The original logs require x+3>0 and x-1>0, so x>1. The value -1+2√3≈2.46 is valid; -1-2√3 is rejected. Answer: x=-1+2√3.
Traps
Forgetting domain restrictions after combining logs; treating Δ<0 as no roots rather than no real roots; sign errors in sum and product; cancelling terms before factorising; rationalising only the numerator.