| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2019 |
| Marks | 5 |
| Topic | Factor & Remainder Theorem |
| Type | Given factor, find all roots |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring factorization after verifying x=1 is a root, then analyzing when the quadratic factor has a repeated root. The verification is trivial substitution, and finding when a quadratic has equal roots using the discriminant is standard A-level technique with minimal problem-solving required. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
Show that, for any value of the real constant $b$, the equation $x^3 - (b + 1)x + b = 0$ has $x = 1$ as a solution.
Find all values of $b$ for which this equation has exactly two real solutions
[5]
\hfill \mbox{\textit{SPS SPS FM 2019 Q10 [5]}}