Hi, I'm 13 years old and I'm from India. I just found out about the forum. Looking a some posts over here, I see that you guys are uber geniuses and I thought that you could provide me a helping hand.

1) If x+y+z=0, show that x³+y³+z³ = 3xyz.
2) If a+b+c=6, a²+b²+c²=14 and a³+b³+c³=26, then find the value of abc.
3) Find x³-1/x³ if x-1/x=a
4) Find x³+1/x³ if x+1/x=4
5) Find x³+1/x³ if x+1/x=√3 (That last sign is a root sign)
6) If x+1/x=p then show that x³+1/x³=p(p²-3)

These are only 6 sums out of about 11 or so. I don't like posting so many either but I can't understand anything. Like in say the 3rd sum, I've been taught how do you solve if the ³ was replaced by a ².

P.S. I'm yet learning the code the PHP or whatever code, so please forgive me for the way I'm posting.

Edit: I thought it'd be better if I posted in a more decent manner now. Thank you guys!

2. Originally Posted by Ruler of Hell
1) If x+y+z=0, show that x³+y³+z³ = 3xyz.
Thus,
$\displaystyle x+y=-z$
Cube both sides,
$\displaystyle (x+y)^3=(-z)^3$
Expand the left side and redo the right,
$\displaystyle x^3+3x^2y+3xy^2+y^3=-z^3$
Thus,
$\displaystyle (x^3+y^3+z^3)+3xy(x+y)=0$
But,
$\displaystyle x+y=-z$
Thus,
$\displaystyle (x^3+y^3+z^3)+3xy(-z)=0$
Thus,
$\displaystyle (x^3+y^3+z^3)-3xyz=0$
Finally,
$\displaystyle x^3+y^3+z^3=3xyz$

3. I so love you! Seriously man you rock!!!!!!!!! Thanks. One sum is better than nothing. My father taught me this sum but he never showed the sum whereby you have to replace the (x+y) with -z. Listen, can you please come on MSN or Gmail? Tell me your id or anything! Please! I need lots of help! *Writes the sum down*
Btw, my mother has told you thanks for helping me!
*Hopes that ThePerfectHacker will come for chatting*

4. Originally Posted by Ruler of Hell
3) Find x³-1/x³ if x-1/x=a
$\displaystyle a^3=(x-1/x)^3=x^3-3x^2/x+3x/x^2-1/x^3$
$\displaystyle =x^3-1/x^3-3(x-1/x)$

$\displaystyle =x^3-1/x^3 -3a$

So:

$\displaystyle x^3-1/x^3=a^3+3a$

RonL

You should check this carefully, I'm not on form in the accuracy
department at the moment.

5. Thanks, that helped me solve 4 sums! Yay! I'll post some more now...

Okay, I'm stuck on this part of a sum.
x³+1/x³ +3√3 = √3 ³

Now what?? Here's the actual sum.

If x+1/x = √3
Find x³+1/x³

Another sum x+1/x=p
Show that x³+1/x³=p(p²+3)
I don't see the reason for writing it in a weird manner. Can I write this at the last step:-
p³+3p and then change it to
p(p²+3)

So this is where I'm stuck. Thanks again. This forum rocks!!!!

6. Hello, Ruler of Hell!

$\displaystyle \text{5) Find }x^3 + \frac{1}{x^3} \text{ if }x + \frac{1}{x}\:=\:\sqrt{3}$

We are given: .$\displaystyle x + \frac{1}{x} \:=\:\sqrt{3}$ .*

Cube both sides: .$\displaystyle \left(x + \frac{1}{x}\right)^3\:=\\sqrt{3})^3$

And we have: .$\displaystyle x^3 + 3x + \frac{3}{x} + \frac{1}{x^3} \;= \; 3\sqrt{3}$

Regroup the terms: .$\displaystyle \left(x^3 + \frac{1}{x^3}\right) + 3\underbrace{\left(x + \frac{1}{x}\right)} \;= \; 3\sqrt{3}$
. . . . . . . . . . . . . . . . . . . . . . . . This is$\displaystyle \sqrt{3}$ *

And we have: .$\displaystyle \left(x^3 + \frac{1}{x^3}\right) + 3\sqrt{3} \;= \;3\sqrt{3}$

Therefore: .$\displaystyle x^3 + \frac{1}{x^3} \;= \;0$

$\displaystyle \text{6) If }x + \frac{1}{x}\,=\,p\text{, then show that: }x^3 + \frac{1}{x^3}\:=\(p^2 - 3)$

Use the same procedure . . .

We are given: .$\displaystyle x + \frac{1}{x} \;= \;p$

Cube: .$\displaystyle x^3 + 3x + \frac{3}{x} + \frac{1}{x^3} \;= \; p^3$

Regroup: .$\displaystyle \left(x^3 + \frac{1}{x^3}\right) + 3\underbrace{\left(x + \frac{1}{x}\right)}\;=\;p^3$
. . . . . . . . . . . . . . . . . This is $\displaystyle p$

And we have: .$\displaystyle \left(x^3 + \frac{1}{x^3}\right) + 3p \;= \;p^3$

Therefore: .$\displaystyle x^3 + \frac{1}{x^3} \;= \; p^3 - 3p \;= \;p(p^2 - 3)$

7. Thanks a lot Soroban. I think I have only another sum or two which I don't know and then I can submit the worksheet!

$\displaystyle \text{1) If }x^2 + \frac{1}{25x^2} = \frac{43}{5} \text{ then find the value of }x^3 + \frac{1}{125x^3}$

Find a³ + b³ + c³ - 3abc
......ab+bc+ac -a² - b² - c²

When, a=-5 b=-6 c=10

That whole thing is a fraction, it wouldn't let me type it as a code. Some [LaTeX Error: Couldn't copy temporary file]

Those are the only two sums I can't figure out. Both of them I've gotten half way through but I can't get a step or two in between. *Gives everyone a bear hug for helping* Oh, and yet again I wish to say Thank You! My grandmother is also happy that I'm seeking help from you guys and my mother will be really delighted once I tell her about you guys. Thanks!

8. Hello, Ruler of Hell!

$\displaystyle \text{1) If }x^2 + \frac{1}{25x^2} = \frac{43}{5} \text{ then find the value of }x^3 + \frac{1}{125x^3}$

This one required Olympic-level gymnastics . . .

We have: .$\displaystyle x^2 + \frac{1}{25x^2}\:=\;\frac{43}{5}$

Add $\displaystyle \frac{2}{5}$ to both sides: .$\displaystyle x^2 + \frac{2}{5} + \frac{1}{25x^2} \;=\;\frac{43}{5} + \frac{2}{5}\;=\;9$

. . and we have: .$\displaystyle \left(x + \frac{1}{5x}\right)^2\;=\;9\quad\Rightarrow\quad \boxed{x + \frac{1}{5x}\;=\;3}$

Cube both sides: .$\displaystyle x^3 + \frac{3x}{5} + \frac{3}{25x} + \frac{1}{125x^3} \;= \;27$

Regroup: .$\displaystyle \left(x^3 + \frac{1}{125x^3}\right) +\frac{3}{5}\underbrace{\left(x + \frac{1}{5x}\right)} \;= \;27$
. . . . . . . . . . . . . . . . . . . This is $\displaystyle 3$

. . so we have: .$\displaystyle \left(x^3 + \frac{1}{125x^3}\right) + \frac{3}{5}(3)\;=\;27\quad\Rightarrow\quad x^3 + \frac{1}{125x^3} \;= \;27 - \frac{9}{5}$

Therefore: .$\displaystyle \boxed{x^3 + \frac{1}{125x^3} \;= \;\frac{126}{5}}$

9. Woohoo yeah! Now one last sum to go! I love you guys!
I've already posted it previously! You guys are so cool! Now I never need to fear Math again. Thanks for teaching me!!!! *Waits for somebody to solve the last sum.... *

10. Hello, Ruler of Hell!

I finally got #2 . . .
I assume there is a typo . . . The sum of cubes should be 36, right?

2) If $\displaystyle \begin{Bmatrix} [1]\;a+b+c=6 \\ [2]\;a^2+b^2+c^2=14 \\ [3]\;a^3+b^3+c^3=36\end{Bmatrix}$, then find the value of $\displaystyle abc.$

From $\displaystyle [1]\;a + b + c \,=\,6$, we have: .$\displaystyle \begin{Bmatrix}a + b = 6 - c \\ b + c = 6 - a \\ a + c = 6 - b\end{Bmatrix}$ (a)

Square $\displaystyle [1]:\;\;(a + b + c)^2\:=\:6^2$

We have: .$\displaystyle \underbrace{a^2 + b^2 + c^2} + \:2(ab + bc + ac) \:=\:36$
From $\displaystyle [2]$, this is $\displaystyle 14$

Then: .$\displaystyle 14 + 2(ab + bc + ac) \:= \:36\quad\Rightarrow\quad ab + bc + ac\:=\:11$ (b)

Cube $\displaystyle [1]:\;\;(a + b + c)^3\:=\:6^3$

We have: .$\displaystyle \underbrace{a^3 + b^3 + c^3} + \:3(a^2b + ab^2 + b^2c + bc^2 + a^2c + ac^2) + 6abc$ $\displaystyle \;= \;216$
From $\displaystyle [3]$, this is $\displaystyle 36$

Then: .$\displaystyle 36 + 3(a^2b + ab^2 + b^2c + bc^2 + a^2c + ac^2) + 6abc$ $\displaystyle \;=\;216$

. . and we have: .$\displaystyle a^2b + ab^2 + b^2c + bc^2 + a^2c + ac^2 + 2abc\;=\;60$

Factor: . . .$\displaystyle ab\underbrace{(a + b)} +\, bc\underbrace{(b + c)} + \,ac\underbrace{(a + c)} + \,2abc\;=\;60$
From (a): .$\displaystyle ab(6 - c) + bc(6 - a) + ac(6 - b) + 2abc \;= \;60$

We have: .$\displaystyle 6ab - abc + 6bc - abc + 6ac - abc + 2abc \;= \;60$

Simplify: .$\displaystyle 6\underbrace{(ab + bc + ac)} -\: abc\;=\;60$
From (b), this is $\displaystyle 11$

Therefore: .$\displaystyle 6(11) - abc\;=\;60\quad\Rightarrow\quad \boxed{abc = 6}$

11. Yay thanks! Yeah there was a typo... Phew. Um.... but um... I.... am sorry but I have another sum here... Err... I don't know what to say... :-(

The thing is a fraction, I couldn't type the code so I used the Underline tag to represent it as a fraction. I'm sorry.. Thats my last sum. If I post another single sum, slap me on the head. Sorry once again (especially to Soroban).

The sum is here:-
Find a³ + b³ + c³ - 3abc
......ab+bc+ac -a² - b² - c²

12. Originally Posted by Ruler of Hell
Yay thanks! Yeah there was a typo... Phew. Um.... but um... I.... am sorry but I have another sum here... Err... I don't know what to say... :-(

The thing is a fraction, I couldn't type the code so I used the Underline tag to represent it as a fraction. I'm sorry.. Thats my last sum. If I post another single sum, slap me on the head. Sorry once again (especially to Soroban).

The sum is here:-
Find a³ + b³ + c³ - 3abc
......ab+bc+ac -a² - b² - c²
$\displaystyle \frac{a^3+b^3+c^3-3abc}{ab+bc+ac-a^2-b^2-c^2}$

Just click on the expression and a box will come up showing you how to code it.

-Dan

13. Hello, Ruler of Hell!

That last one seems to just be an arithmetic problem . . .

Find .$\displaystyle \frac{a^3 + b^3 + c^3 - 3abc}{ab + bc + ac -a^2 - b^2 - c^2}$ .when $\displaystyle a = -5,\; b = -6,\;c=10$

We have: .$\displaystyle \frac{(-5)^3 + (-6)^3 + 10^3 - 3(-5)(-6)(10)}{(-5)(-6) + (-6)(10) + (-5)(10) - (-5)^2 - (-6)^2 - 10^2}$

. . . . . $\displaystyle = \;\frac{-125 - 216 + 1000 - 900}{30 - 60 - 50 - 25 - 36 - 100}$ $\displaystyle =\;\frac{-241}{-241}\;= \;\boxed{1}$

The numerator just happens to equal the denominator?

Well, there is a reason, but it depends on the values of $\displaystyle a,\,b,\,c$
. . . . . and my proof is truly awful!

14. I got the exactly same thing, though I didn't directly substitute. First I cancelled the squares and cubes then I did the stuff. First I got zero, then I got a 1. I got afraid and so I left it out. Thanks Soroban! I owe you quite a lot! Yay, now to finish the final sum!!!! Hurrah! What joy! Sorry if I wasted quite a lot of time of yours... and anybody else...

15. Hello, Ruler of Hell!

I got the exactly same thing, though I didn't directly substitute.
First I cancelled the squares and cubes . . . No !!
Shame on you . . . you know that's illegal!
Okay, here's my clumsy proof . . .

Since $\displaystyle a = -5,\;b = -6,\;c = 10$, we have: $\displaystyle a + b + c \:=\:-1$

. . Hence: .$\displaystyle a + b \:=\:-c -1$ [1]

Cube [1]: .$\displaystyle (a + b)^3\;=\;(-c-1)^3$

. . . $\displaystyle a^3$$\displaystyle + 3a^2b + 3ab^2 + b^3\;=\;-c^3 - 3c^2 - 3c - 1 Arrange terms: .\displaystyle a^3 + b^3 + c^3 + 3a^b + 3ab^2 \;= \;-3c^2 - 3c - 1 Factor: .\displaystyle a^3 + b^3 + c^3 + 3ab\underbrace{(a + b)} \;= \;-3c^2 - 3c - 1 . . . . .\displaystyle a^3 + b^2 + c^3 + 3ab(-c - 1) \;=\;-3c^2 - 3x - 1 . . . . . .\displaystyle a^3 + b^3 + c^3 - 3abc - 3ab \;= \;-3c^2 - 3c - 1 . . . . . . . . . . . \displaystyle a^3 + b^3 + c^3 - 3ab \;= \;3ab - 3c^2 - 3c - 1 Hence: .\displaystyle \boxed{Numerator \;= \;3ab - 3c^2 - 3c - 1} Square [1]: .\displaystyle (-c-1)^2\;=\;(a + b)^2 . . . . . . . .\displaystyle c^2 + 2c + 1 \;= \;a^2 + 2ab + b^2 . . . . . . . . . \displaystyle -a^2 - b^2 \;= \; 2ab - c^2 - 2c - 1 Add \displaystyle ab + bc + ac - c^2 to both sides: . . . \displaystyle (ab + bc + ac - c^2) - a^2 - b^2 \;= \;(ab + bc + ac - c^2)$$\displaystyle + 2ab - c^2 - 2c - 1$

. . . . $\displaystyle ab + bc + ac - a^2 - b^2 - c^2 \;= \;3ab + ac + bc - 2c^2 - 2c - 1$

We have: .$\displaystyle Denominator \;= \;3ab + c\underbrace{(a + b)} - 2c^2 - 2c - 1$

. . . . . . . . . . . . . . . . . . . $\displaystyle = \;3ab + c(-c - 1) - 2c^2 - 2c - 1$

. . . . . . . . . . . . . . . . . . . $\displaystyle = \;3ab - c^2 - c - 2c^2 - 2c - 1$

Hence: .$\displaystyle \boxed{Denominator \;= \;3ab - 3c^2 - 3c - 1}\quad\hdots\quad QED\,!$
.

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