Algebra - IOQM Study Guide
Algebra on IOQM tests structural manipulation rather than long computation: polynomials, systems, inequalities, sequences, and functional relations. The exam expects a compact identity, substitution, or inequality that turns the problem into one or two lines.
Core rules and concepts
- For symmetric variables set p=a+b+c, q=ab+bc+ca, r=abc. Then a²+b²+c²=p²−2q and a³+b³+c³=p³−3pq+3r.
- Vieta gives sums and products of roots; match signs exactly.
- AM–GM and Cauchy–Schwarz need nonnegative terms and equality checks.
- For sums, write the general term and look for telescoping or a recurrence.
- Record the domain before cancelling: denominators, radicals, logarithms, and integer constraints.
Worked example
If a+b+c=0 and a²+b²+c²=1, find 100(a⁴+b⁴+c⁴). First q=ab+bc+ca=((a+b+c)²−(a²+b²+c²))/2=−1/2. Now 1=(a²+b²+c²)²=a⁴+b⁴+c⁴+2(a²b²+b²c²+c²a²). Also a²b²+b²c²+c²a²=(ab+bc+ca)²−2abc(a+b+c)=q²=1/4. So a⁴+b⁴+c⁴=1−2(1/4)=1/2. Multiplying by 100 gives 50, the two-digit answer.
Traps that cost marks
- Squaring both sides without checking extraneous roots.
- Cancelling a factor that can be zero.
- Applying AM–GM to terms that may be negative.
- Mixing Vieta signs, especially for cubics.
- Ignoring that variables are integers or positive when stated.