Using sum/product of roots of a cubic equation to solve given expression $(a+b)^3+ (b+c)^3 + (c+a)^3$

Question

If $a, b, c$ are the roots of the equation $7x^3- 25x +42 =0$, then the value of the expression $(a+b)^3+ (b+c)^3 + (c+a)^3$ is?

I tried to solve this but wasn't able to simplify the term to be able to apply sum and product of roots of equations. Could someone help me through simplification of the expression to make this solvable?

Since $a+b+c = \frac{-25}{7}$, $ab + bc + ca = 0$ and $abc = -42$, if i factorize the expression i get

$(2[(a+b+c)^3 - 3a^2(b+c) - 3 c^2 (a+b) -3b^2(a+c) -6abc]) - 3(a+b)^2(b+2c+a) - 3(c+a)^2(c+2b+a) - 3(b+c)^2(c+2a+b))$

This is what i get, after my best attempts at simplification. Since this is an mcq type question, i believe this does not require such mammoth efforts. Could someone provide me a hint and possibly an easier way of simplification/application of the concept?

score 2 · Accepted Answer · edited Apr 13 '17 at 12:20

2

Using Vieta's formula $\displaystyle a+b+c=0,(-1)^3abc=42$

$$\implies \sum (a+b)^3=-\sum c^3$$

Method $\#1:$

As $a,b,c$ individually satisfy the given equation, we can write $$7a^3=25a-42\text{ etc.}$$ Again using Newton's Power Sum formula $$7\sum a^3=25\sum a-3\cdot42=\cdots$$

Method $\#2:$ Like If $a,b,c \in R$ are distinct, then $-a^3-b^3-c^3+3abc \neq 0$.,

$$a^3+b^3+c^3=3abc$$

edited Apr 13 '17 at 12:20

Community

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answered Aug 23 '14 at 16:34

lab bhattacharjee

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I apologize since i am not familiar with those formulas and their applications, could you suggest perhaps, a more algebraic manipulation oriented method? – Mathbreaker Aug 23 '14 at 16:38
@Mathbreaker, How have you written $a+b+c=\dfrac{25}7?$ – lab bhattacharjee Aug 23 '14 at 16:39
I apologize, it should be a+b+c = 0, and ab+bc+ca = 25/7, right? – Mathbreaker Aug 23 '14 at 16:41
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@Mathbreaker, $$ab+bc+ca=-(-1)^2\dfrac{25}7. (-1)^3abc=42$$ – lab bhattacharjee Aug 23 '14 at 16:42
I don't think the Power Sum is necessary in the last step. – Quang Hoang Aug 23 '14 at 16:42
Is there any way the expression could be simplified so that i can use the given information to find its value? – Mathbreaker Aug 23 '14 at 16:44
@QuangHoang, Added alternative method – lab bhattacharjee Aug 23 '14 at 16:45
@Mathbreaker, Please find the edited version – lab bhattacharjee Aug 23 '14 at 16:46
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@labbhattacharjee: I meant the words "Power Sum" obviously scared Mathbreaker. Actually you just used the fact that $a,b,c$ are roots of the original polynomial in the last step. – Quang Hoang Aug 23 '14 at 16:48
Thanks that helped, however, even though the first part is now simplified to some extent, i fail to see how $3(a+b)^2(b+2c+a) - 3(c+a)^2(c+2b+a) - 3(b+c)^2(c+2a+b))$ could further be simplified. Apologies, i do not have much insight on simplifications such as these... – Mathbreaker Aug 23 '14 at 16:49
@QuangHoang, Agreed & adjusted accordingly. Thanks for your input – lab bhattacharjee Aug 23 '14 at 16:52

Using sum/product of roots of a cubic equation to solve given expression $(a+b)^3+ (b+c)^3 + (c+a)^3$

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