Standard +0.3 This is a straightforward application of Vieta's formulas with given root relationships. Students must set up three equations from sum and product of roots, then solve the resulting system. While it requires careful algebra across multiple steps, the method is standard for FP1 and involves no conceptual difficulty beyond applying well-practiced techniques.
5 The cubic equation \(x ^ { 3 } - 5 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , - 3 \alpha\) and \(\alpha + 3\). Find the values of \(\alpha , p\) and \(q\).
Matching other coefficients; one mark for each, ft incorrect \(\alpha\)
# Question 5:
Sum of roots $= \alpha+(-3\alpha)+\alpha+3=3-\alpha=5$ | M1 | Use of sum of roots
$\Rightarrow \alpha=-2$ | A1 |
Product of roots $= -2\times6\times1=-12$ | M1 | Attempt to use product of roots
Product of roots in pairs $= -2\times6+(-2)\times1+6\times1=-8$ | M1 | Attempt to use sum of products of roots in pairs
$\Rightarrow p=-8$ and $q=12$ | A1, A1 [6] | One mark for each, ft if $\alpha$ incorrect
**Alternative:**
$(x-\alpha)(x+3\alpha)(x-\alpha-3)=x^3+(\alpha-3)x^2+(-5\alpha^2-6\alpha)x+3\alpha^3+9\alpha^2$ | M1 | Attempt to multiply factors
$\Rightarrow \alpha=-2$ | M1A1 | Matching coefficient of $x^2$, cao
$p=-8$ and $q=12$ | A1A1 [6] | Matching other coefficients; one mark for each, ft incorrect $\alpha$
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5 The cubic equation $x ^ { 3 } - 5 x ^ { 2 } + p x + q = 0$ has roots $\alpha , - 3 \alpha$ and $\alpha + 3$. Find the values of $\alpha , p$ and $q$.
\hfill \mbox{\textit{OCR MEI FP1 2009 Q5 [6]}}