Standard +0.3 This is a standard C1 factor theorem question requiring students to use the given roots to form equations. Since the curve touches at x=3, students must recognize this means (x-3) is a repeated factor. The algebra involves solving simultaneous equations from f(-1)=0, f(3)=0, and f'(3)=0, which is routine for this topic with clear scaffolding from the 'show that' structure.
\includegraphics{figure_1}
Figure 1 shows the curve with equation \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \((-1, 0)\) and touches the \(x\)-axis at the point \((3, 0)\).
Show that \(a = -5\) and find the values of \(b\) and \(c\). [5]
\includegraphics{figure_1}
Figure 1 shows the curve with equation $y = x^3 + ax^2 + bx + c$, where $a$, $b$ and $c$ are constants. The curve crosses the $x$-axis at the point $(-1, 0)$ and touches the $x$-axis at the point $(3, 0)$.
Show that $a = -5$ and find the values of $b$ and $c$. [5]
\hfill \mbox{\textit{Edexcel C1 Q4 [5]}}