#further maths cambrige quadratic

71 messages · Page 1 of 1 (latest)

plucky harbor
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I just need to match my answers as i dont have the markscheme

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noble deltaBOT
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plucky harbor
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Cubed both sides to get this equation, substituted y, and plugged in S2 formula

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I dont know if its correct

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Doesnt match with the answer in my notes

sage kite
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Hm. Not sure how that substitution helps.
I guess you can try expressing S(6) as some composition of symmetric polynomials. Though, I can't say I see an easy way to do that right away.

plucky harbor
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It should work

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Took even powers to other side and made the whole powers and all a factor of 3 to substitute in y

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Shouldnt S2 (in terms of y) equal to S6 in terms of x?

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After all the question asked to use the y substitution

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I just need to know the answer

sage kite
plucky harbor
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X3 + 1 lhs
-5x2 rhs

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Cube both sides

sage kite
rigid girderBOT
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Couldn't find an attached image in the last 10 messages.

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Couldn't find an attached image in the last 10 messages.

plucky harbor
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Here

thick vector
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guess it doesnt work

plucky harbor
thick vector
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thanks

sage kite
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Ahh, I see. Clever approach!
So, now we get y1^2 + y2^2 + y3^2 = (y1 + y2 + y3)^2 - 2(y1y2 + y1y3 + y2y3). And by Vieta's formulas we have y1 + y2 + y3 = -128, y1y2 + y1y3 + y2y3 = 3.
On the other hand, since y = x^3, that is also equal to x1^6 + x2^6 + x3^6.

plucky harbor
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Wanted to just straight up add the roots with 6th power but all roots except for one was complex

thick vector
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well we have 1 real root and 2 complex roots

plucky harbor
sage kite
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So, that's correct.

plucky harbor
sage kite
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You don't need to find the roots.

plucky harbor
sage kite
plucky harbor
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Did Ea -2Eab

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E is summation symbol

sage kite
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Yeah. That's what we both wrote.

plucky harbor
plucky harbor
thick vector
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a + b + c = -5,
ab + bc + ca = 0,
abc = -1.

plucky harbor
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Ye

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Checks out

thick vector
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just do the manipulations and substitutions for abc

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but lets have a look at the expression a+b+c = -5

plucky harbor
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Ok

thick vector
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if we raise it to the 6th we end up with out expression a^6+b^6+c^6 with more terms but we notice we find our first step to finding a^6+b^6+c^6

plucky harbor
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Yes

thick vector
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but thats going to be a lot of work

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and expanding

plucky harbor
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But we have to do binomial expansion

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And algebra

thick vector
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yeah and its a pain

plucky harbor
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A lot of it

thick vector
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thats one way i though of going about it

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but not if we manipulate our expression

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we dont want to do (a+b+c)^6

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we can do (a+b)^6=(-5-c)^6

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so that it makes our life easier than to expand the sum of three variables

plucky harbor
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One thing i dont fully understand is that how does S’2=S6, i know that y=x^3 but still idk the indepth explanation

plucky harbor
sage kite
thick vector
sage kite
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Again, expressing the initial sum in terms of symmetric polynomials is possible, but it will take a very long time.

thick vector
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but sometimes to avoid the long extra work you just learn how to make clever maniputions to make the work easier

sage kite
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Well, I mean, the recommended substitution is already given in the statement.

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I'm pretty sure Cacharo's approach is the intended one.

thick vector
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it seems that way, i just gave my approach to the problem

plucky harbor
thick vector
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but your approach works as well

plucky harbor
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Thanks guys

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Appreciate it