Puzzles
24 December
The expression \((3x-1)^2\) can be expanded to give \(9x^2-6x+1\). The
sum of the coefficients in this expansion is \(9-6+1=4\).
What is the sum of the coefficients in the expansion of \((3x-1)^7\)?
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The sum of the coefficients can be worked out by substituting \(x=1\) into the polynomial, so the sum of the coefficients is \((3-1)^7\), or 128.
8 December
The equation \(x^5 - 7x^4 - 27x^3 + 175x^2 + 218x = 840\) has five real solutions. What is the product of all these solutions?
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The product of all the solutions is 840.
Expanding \((x-\alpha)(x-\beta)(x-\gamma)(x-\delta)(x-\epsilon)=0\) gives \(x^5 + \dots = \alpha\beta\gamma\delta\epsilon\). As long as the
coefficient of \(x^5\) is 1, the constant term when written like this will be the product of all the solutions (for this to always work, you will need to include repeated solutions and complex solutions.)
Extension
The equation \(x^8 - 19x^7 + 126x^6 - 294x^5 - 231x^4 + 1869x^3 - 1576x^2 - 1556x + 1680=0\) has eight real solutions. What is the product of all these solutions?
