Polynomials

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Student 999
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#1
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Is my working out adequate for parts i and ii? Also for the last part I'm not sure how to approach it, I kind of followed back to part I and made an equation that is exists only when the roots are in arithmetic progression am I missing other steps?Name:  Screenshot 2022-07-01 at 11.07.25.png
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mqb2766
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I guess Id have posed iib) a bit differently as youve assumed its geometric and showed its ok. More generally three roots could be
r, ry and rz
for two arbitrary multipliers y, z. Then yz = 1, so ... It may just be the way youve written it as youre saying the product of the other two roots is r^2, so then its just justifying why the corresponding multipliers cancel when you multply them.

Last part, you seem to have shown one way only (looks about right, but not carefully checked). If r, p, q satisfy that relationship, can you then show the roots are in AP. THis is essentially the two directions/approaches in i) and ii) for a geometric relationship.
Last edited by mqb2766; 1 month ago
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Student 999
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#3
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(Original post by mqb2766)
I guess Id have posed iib) a bit differently as youve assumed its geometric and showed its ok. More generally three roots could be
r, ry and rz
for two arbitrary multipliers y, z. Then yz = 1, so ... It may just be the way youve written it as youre saying the product of the other two roots is r^2, so then its just justifying why the corresponding multipliers cancel when you multply them.

Last part, you seem to have shown one way only (looks about right, but not carefully checked). If r, p, q satisfy that relationship, can you then show the roots are in AP. THis is essentially the two directions/approaches in i) and ii) for a geometric relationship.
I see would it simply be to state that p/3 = Is a root, then comment that p/3 -d and p/3 +d are also roots hence why the sum of roots is 3a since the common difference cancels out? Resulting to my above equation to be a sufficient condition
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mqb2766
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(Original post by Student 999)
I see would it simply be to state that p/3 = Is a root, then comment that p/3 -d and p/3 +d are also roots hence why the sum of roots is 3a since the common difference cancels out? Resulting to my above equation to be a sufficient condition
Youve shown the necessary condition on p,q,r given that the roots are in arithmetic sequence, which is essentially part i). You want to show that given pq-3r=... condition, then the roots will be in an arithmetic sequence (sufficient) which is essentially part ii). So follow a similar argument to part ii).
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Student 999
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(Original post by mqb2766)
Youve shown the necessary condition on p,q,r given that the roots are in arithmetic sequence, which is essentially part i). You want to show that given pq-3r=... condition, then the roots will be in an arithmetic sequence (sufficient) which is essentially part ii). So follow a similar argument to part ii).
My bad I see thanks
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