Find unknown coefficients from roots

Questions providing some roots and asking students to find unknown coefficients in the polynomial by using factor theorem or expanding and comparing coefficients.

2 questions · Standard +0.3

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Pre-U Pre-U 9794/2 Specimen Q7
12 marks Standard +0.3
7 A cubic polynomial is given by $$\mathrm { P } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
  1. If \(\mathrm { P } ( x )\) is exactly divisible by \(x - 1\), and has a local maximum at \(x = - 1\), determine the values of \(a\) and \(b\).
  2. Sketch the curve \(y = \mathrm { P } ( x )\), marking the intercepts and the \(x\)-coordinates of the stationary points.
  3. Expand and simplify \(\mathrm { P } ( 1 + x )\), and deduce that \(\mathrm { P } ( 1 + x ) = - \mathrm { P } ( 1 - x )\). Interpret this result graphically.
Edexcel C2 Q3
7 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve \(y = f(x)\) where $$f(x) = 4 + 5x + kx^2 - 2x^3,$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A\), \(B\) and \(C\). Given that \(A\) has coordinates \((-4, 0)\),
  1. show that \(k = -7\), [2]
  2. find the coordinates of \(B\) and \(C\). [5]