Polynomials and Quadratic Equations
This domain tests the link between zeroes and coefficients, the discriminant, and translating word problems into quadratics. Expect one relation-of-roots question, one nature-of-roots question, and one applied problem.
Core rules
- For ax² + bx + c (a ≠ 0) with zeroes α, β: α + β = −b/a and αβ = c/a.
- For a cubic ax³ + bx² + cx + d: sum = −b/a, sum of products taken two at a time = c/a, product = −d/a.
- Reconstructing a quadratic from its zeroes: k[x² − (α + β)x + αβ], k ≠ 0.
- Discriminant D = b² − 4ac. D > 0 gives two distinct real roots, D = 0 gives equal roots, D < 0 gives no real roots.
- Roots: x = (−b ± √D)/(2a). Completing the square writes ax² + bx + c as a(x + b/2a)² + (c − b²/4a).
Worked example
If α and β are the zeroes of x² − 5x + 6, find α² + β² and 1/α + 1/β without solving. Here α + β = 5 and αβ = 6 (note the sign change from −b/a). α² + β² = (α + β)² − 2αβ = 25 − 12 = 13. 1/α + 1/β = (α + β)/(αβ) = 5/6. The roots happen to be 2 and 3, and both results check.
Traps
- Dropping the minus sign in −b/a; the quadratic above has positive zeroes because b is negative.
- Writing the sum of roots as b/a or the product as −c/a.
- Dividing by αβ when it may be zero.
- Reporting D < 0 as “no solution” when the equation has complex, not real, roots.
- In area and age problems, keeping a negative or physically impossible root instead of rejecting it with a reason.