Algebra: Polynomials, Linear and Quadratic Equations
Algebra: Polynomials, Linear and Quadratic Equations is the computational backbone of the paper. The exam tests remainder and factor theorems, zeros and coefficient relations, consistency of linear pairs, and discrimination between real, equal and non-real roots.
Core rules
- Remainder theorem: p(x) divided by (x−a) leaves p(a); (x−a) is a factor iff p(a)=0.
- For ax²+bx+c=0, discriminant D=b²−4ac: D>0 two distinct real roots, D=0 equal real roots, D<0 no real roots.
- Roots satisfy α+β=−b/a and αβ=c/a; x²−(α+β)x+αβ=0.
- Linear pair a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0: unique solution if a₁/a₂≠b₁/b₂; infinitely many if a₁/a₂=b₁/b₂=c₁/c₂; none otherwise.
- Quadratic formula: x=[−b±√D]/(2a).
Worked example
Solve 2x²−7x+3=0. D=49−24=25. x=(7±5)/4 gives x=3 or x=1/2. Check: sum 3.5=−(−7)/2, product 1.5=3/2.
Traps
Do not cancel coefficients across an equation without preserving the equation. A zero discriminant means one repeated root, not no root. When a pair is dependent, do not report a unique solution. In word problems, reject roots that violate the context, such as a negative length.