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Math people, could you help us with this hard Gelfand's factoring problem?


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It's from Gelfand's. We have Charon's solutions, and that helped in the one just before this, but she had trouble reading the notation (with the ^ for the powers, etc) and still doesn't get it. Also, Charon wrote, "Holy cow? This one was a toughie! The only reason I evntually guessed the right answer was because I started to look ahead..." I'll

 

Problem 122

 

d) a^3 + b^3 + c^3 -3abc

 

His hint is that (a+b+c) is probably a factor. The way it came up in his pdf file was hard for her to read. She needs to understand this to move onto e which is:

 

3) (a+b+c)^3 -a^3-b^3-c^3

 

Help on that one might be good, too, even though he has a solution, but mostly help on d.

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It's from Gelfand's. We have Charon's solutions, and that helped in the one just before this, but she had trouble reading the notation (with the ^ for the powers, etc) and still doesn't get it. Also, Charon wrote, "Holy cow? This one was a toughie! The only reason I evntually guessed the right answer was because I started to look ahead..." I'll

 

Problem 122

 

d) a^3 + b^3 + c^3 -3abc

 

His hint is that (a+b+c) is probably a factor. The way it came up in his pdf file was hard for her to read. She needs to understand this to move onto e which is:

 

3) (a+b+c)^3 -a^3-b^3-c^3

 

Help on that one might be good, too, even though he has a solution, but mostly help on d.

 

Answer for d: Yes, factor it.

You should get:

(a+b+c)(a^2+b^2+c^2-ab-ac-bc).

 

ETA: The way I got the answer was using long division and looking for patterns of what the products would be and considering the signs of the addends. It was messy, so in this type of problem it helps to stay focused and organized in how you write out your answer. Carefully checking your work helps, too. I have not seen Charon's solutions, so I am not sure if he wrote out the steps to solve it.

Edited by fractalgal
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Okay, just divide. I'll tell her that. Sounds so much more simple than the way Charon explained it on the page. I think she tried it, but I'll have her take another stab at it. The world won't come to an end if she doesn't get it (or will it???), but she needs to do understand it in order to succeed with the next one.

 

So, does this all look fairly straightforward to you? If you'd seen this without the hint, would it have been challenging?

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Heck yes!

 

 

Thank you. What was confusing was when Charon wrote it out, things didn't fit neatly on one line, which with the ^, etc was hard for her to understand. If she still can't get this, we'll set aside and come back to it later (she is at what Jane Austin calls "That trying age" in Pride and Prejudice.) She's in what she considers a very easy part of Dolciani (seeing as she has done most of Lial's) so that helps. Previously, Gelfand's was the easier one. I choose whichever is easier for days when she is a walking stem...

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Karin,

It would be a challenge to copy my division into this message block, but, if you need a copy of my work, I could scan and email it.

 

Just let me know.

 

Jane

 

I'll have dd try it again first. If not...maybe I'll take you up on it. What would I do without this forum? A ps hs girl who works at our local library told me that I helped her with her college stuff more in 5 minutes than her counsellor has all year. With things I learned here after asking for some research suggestions for dd's highschool next year.

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So, does this all look fairly straightforward to you? If you'd seen this without the hint, would it have been challenging?

 

If I'd not seen it before, yes I think it would be challenging; although if I knew it could be factored, (a+b+c) would be one of the first factors to be considered.

 

This factorization for d. is useful in proving the Arithmetic Mean Geometrical Mean inequality for n numbers....particularly the case n=3 which shows that the geometric mean does not exceed the arithmetic mean. I have seen this before in my study of Real Analysis. This theorem is likely coming up in her book if the author is having her learn this now.

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This factorization for d. is useful in proving the Arithmetic Mean Geometrical Mean inequality for n numbers....particularly the case n=3 which shows that the geometric mean does not exceed the arithmetic mean. I have seen this before in my study of Real Analysis. This theorem is likely coming up in her book if the author is having her learn this now.

 

 

Thanks for this information--I'll check that out.

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:party: She understands it now!!! Thanks so much for your help.

 

However, in the interest of full disclosure, she was so opposed to solving it with long division, the way you did here for me, she went back to Charon's sheet and re-examined on her own and understood it that way. Not that she's ever contrary ;), but she is 13.

 

But it wouldn't have happened without your help, so please don't stop helping me. It gave me great and wonderful leverage, plus I can keep this information for my other dc who may do it by division. The link to Charon's PDF file on the solutions for Gelfands is at the Drat Those Greeks Blog--he moved it a few times, so I'm not sure if I have the right one in my favourites. http://myrtlehocklemeier.blogspot.com is the link to that blog, and the link is in the right column.

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