STEP - Question Help - Cambridge Maths

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Hi, I'm working through STEP orientated worksheets (not the actual past papers) in preparation for STEP I/II exams and I've came across a question in which I'm completely stumped and spent hours trying to reach a solution. Any help/solutions would be greatly appreciated, thanks a lot.

Knowing that $\alpha_1,\alpha_2,\alpha_3$ are the three roots of the polynomial, what three equations can you write down about the relationships between the roots, based on the coefficients of the polynomial (I'm assuming you've covered roots of polynomials, at least for quadratics).

Then try and rewrite your expressions involving those forms. That will take care of the first two parts.

If you've not covered roots of polynomials, then:



For the final part, $\alpha_1$ is a root of the polynomial, so .... That's only a small hint, but hopefully may be enough.

Hi, I'm working through STEP orientated worksheets (not the actual past papers) in preparation for STEP I/II exams and I've came across a question in which I'm completely stumped and spent hours trying to reach a solution. Any help/solutions would be greatly appreciated, thanks a lot.

I managed to get b) by expanding (a1+a2+a3)^2

, but I'm struggling with a) and c) a bit still... I'm trying the same method for c) as b); expanding (a1+a2+a3)^2.
Any help is majorly appreciated, thanks a lot !!!

Alright i just had a go at the question:

So a1, a2 and a3 are roots.

Therefore: (x-a1)(x-a2)(x-a3) = 5x^3 - x^2 - 2x +3

Multiplying out left side gives:
x^3 - x^2(a1+a2+a3)+(a2a3 + a1a2 + a1a3) - a1a2a3

Since 1/a1 + 1/a2 + 1/a3 can be written as a1a2 + a2a3 + a1a3 / a1a2a3
Then when we let x = 0 on both sides we see a1a2a3 = -3
next equating coefficients of x from both sides
gives a2a3+a1a2+a1a3 = -2

Hence 1/a1 + 1/a2 + 1/a3 = -2/-3 = 2/3

Can you finish the rest?

Thanks a lot for the help!! Greatly appreciated. I tried for c), the way that ghostwalker said to do it

"For c) I'd go with

since is a root of the polynomial, then we know



Similarly for the other two roots.

Now add your three equations."

But I can't seem to get a solution from that method
Can you perhaps share how you would do it? Thanks a lot.

Thanks a lot for the help!! Greatly appreciated. I tried for c), the way that ghostwalker said to do it

"For c) I'd go with

since is a root of the polynomial, then we know



Similarly for the other two roots.

Now add your three equations."

But I can't seem to get a solution from that method
Can you perhaps share how you would do it? Thanks a lot.

were did you get these notes?

what notes?

the OPs notes