Question - Relationship Between the Roots of a Cubic Equation and its Coefficients.

My problem comes from the Further Pure Maths 2 textbook from AQA (UK exam board - A-Level).

Page 28 of the textbook: http://filestore.aqa.org.uk/subjects...2-TEXTBOOK.PDF

Here's the question in the texbook:

http://oi44.tinypic.com/2yn4512.jpg

Now I'm confused by the workings out of question (b), specifically on the first line where it says $\displaystyle (4 \times 3)$. I would have though it would have been just $\displaystyle 4$. Could someone explain to me why the 4 is multiplied by 3? I understand everything else but this does not make sense to me.

Here is some background information on the previous page (page 27 in the textbook):

http://oi43.tinypic.com/8xtp5j.jpg

Re: Question - Relationship Between the Roots of a Cubic Equation and its Coefficient

The problem is the crap notation employed, the term in question is the sum of products of pairs of distinct roots, if the original roots had been $\displaystyle \rho_1,\ \rho_2$ and $\displaystyle \rho_3$ the relevant term would be:

$\displaystyle \sum_{I} (\rho_i-2)(\rho_j-2) = (\rho_1-2)(\rho_2-2)+(\rho_1-2)(\rho_3-2)+(\rho_2-2)(\rho_3-2)$

where $\displaystyle I$ is the index set so the sum is over distinct (unordered) pairs $\displaystyle (i,j)$

Then there are three terms in the sum each with constant term $\displaystyle 4$ which gives $\displaystyle 3 \times 4$.

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Re: Question - Relationship Between the Roots of a Cubic Equation and its Coefficient

In more detail:

$\displaystyle \begin{align*}(\alpha-2)(\beta-2) + (\beta-2)(\gamma-2) + (\alpha-2)(\gamma-2)& =\alpha\beta- 2(\alpha+\beta) +4+ {\,}\\ &\hspace{1.5em}\beta\gamma- 2(\beta+\gamma) +4+{\,}\\& \hspace{1.5em}\alpha\gamma- 2(\alpha+\gamma) +4\\& =\alpha\beta+\beta\gamma+ \alpha\gamma -4(\alpha+\beta+\gamma) +4\times3\end{align*}$

Re: Question - Relationship Between the Roots of a Cubic Equation and its Coefficient

Thank you guys! I actually figured it out this morning out of the blue, but your explanations were great to solidify my understanding. Hopefully the rest of the notation in this textbook isn't as bad! Thanks again.