Algebra
Algebra in SEE Optional Mathematics tests whether you can move fluently between symbolic form and graph/table form. The unit bundles functions, polynomials, sequences and series, quadratic equations, and linear programming; questions often combine two ideas, such as using a quadratic model inside a function problem.
Functions and polynomials
A function is a rule assigning each domain element exactly one range element. For f(x)=..., compute f(a) by substitution; for composite f(g(x)), work inside-out. The remainder theorem says the remainder of p(x) divided by (x-a) is p(a); the factor theorem says (x-a) is a factor exactly when p(a)=0. Use synthetic division or inspection to factor cubics.
Sequences and series
Arithmetic: t_n=a+(n-1)d, S_n=n/2[2a+(n-1)d]. Geometric: t_n=ar^{n-1}, S_n=a(r^n-1)/(r-1), r≠1. For infinite geometric with |r|<1, S∞=a/(1-r). Identify whether the problem asks for a term, number of terms, or sum.
Quadratic equations
For ax²+bx+c=0, x=(-b±√(b²-4ac))/(2a). Discriminant D=b²-4ac: D>0 two real roots, D=0 equal real roots, D<0 no real roots. Sum roots=-b/a, product=c/a.
Linear programming
Define decision variables, write constraints as inequalities, graph the feasible region, evaluate the objective at corner points; the optimum occurs at a vertex for a linear objective.
Worked example
Solve x²-5x+6=0. Factor: (x-2)(x-3)=0, so x=2 or 3. Check: 4-10+6=0 and 9-15+6=0.
Traps
Losing the negative root when taking square roots; forgetting r≠1 in a geometric sum; shading the wrong side of an inequality; evaluating the objective at an interior point rather than at vertices.