Algebra
Algebra at Leaving Certificate Higher Level tests fluent manipulation, equation solving, inequalities, complex numbers, sequences and series, and formal proof. Questions often combine skills, so a correct final answer without justified steps can lose marks.
Core rules
- Factor theorem: ((x-a)) is a factor of (f(x)) exactly when (f(a)=0). Use it to factor cubics and quartics before solving.
- For (ax^2+bx+c=0), the discriminant (b^2-4ac) determines the number of real roots. With real coefficients, non-real roots occur in conjugate pairs.
- For inequalities, solve the corresponding equation first, then test intervals. Multiplying or dividing by a negative reverses the inequality sign.
- Log laws: (\log(ab)=\log a+\log b), (\log(a/b)=\log a-\log b), (\log(a^n)=n\log a), with arguments positive.
- Complex numbers: (i^2=-1); the conjugate of (a+bi) is (a-bi); (|a+bi|=\sqrt{a^2+b^2}). Rationalise denominators using the conjugate.
- Arithmetic sequence: (T_n=a+(n-1)d), (S_n=\frac n2(2a+(n-1)d)). Geometric sequence: (T_n=ar^{n-1}), (S_n=\frac{a(1-r^n)}{1-r}), and (S_\infty=\frac{a}{1-r}) for (|r|<1).
Worked example
Solve (x^2-5x+6<0). Factor: ((x-2)(x-3)<0). Critical values are 2 and 3. Test (x=0): the product is positive, so the expression is negative between the roots. Hence (2<x<3).
Traps
Dividing by a variable and losing a root; forgetting the (\pm) when taking a square root; ignoring log-domain restrictions; reversing an inequality incorrectly; writing the conjugate with the wrong sign; using the geometric sum formula when (r=1).