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Well There are numerous problems, I'll post up a few at a time. This is part of a larger assignment and I am so ashamed to say that I don't have any idea on some of them.
1) Factor [itex]x^8+2x^4y^4+9y^8[/itex]
and [itex]a^4+b^4+c^2 - 2(a^2b^2 +a^2c+b^2c)[/itex]
I thought maybe it was some perfect square, the first one. I noticed the powers were quite coincidental so i let u=x^4 and v=y^4, but it seems there are no numbers that multiply to 9 and add to 2...I think that's irreducible. The second one I hoped I would notice its some expansion of something familiar, but I didnt get it..
Edit: I just tried Mathematicia and get [itex](x^4-2x^2y^2+3y^4)(x^4+2x^2y^2+3y^4) \mbox{and} (a^2-2ab+b^2-c)(a^2+2ab+b^2-c)[/itex] But I want to know how to get them manually!
2) Prove [itex]20^{22}-17^{22} +4^{33} -1[/itex] is divisible is 174.
I think i might need some mod arithmetic or something here, but I haven't done that before. Also, I noticed the differences, 20-17, 4-1, are the same, 3, which 174 is divisible by. And perhaps even more useless, that 174 is a 17 and 4 put together, which are in there...I tried expanding 20^22-17^22 and 4^33 -1 by [itex]x^n-1=(x-1)(1+x+x^2+x^3\cdots +x^{n-2}+x^{n-1})[/itex] and the similar one for x^n-y^n, but no go.
Edit: Once again mathematicia says the actualy number is 40768477197265827835864143774, and when divided by 174 gives 234301593087734642734851401, but I want to know how to do it.
3) Prove that if p,q,r and s are odd integers, then [itex]x^{10}+px^9-qx^7+rx^4-s=0[/itex] has no integer roots.
Just no idea on that one, none at all.
4) For which real values of b do the equations [itex]x^3+bx^2+2bx-1=0 \mbox{and} x^2+(b-1)x+b=0[/itex] have a common root?
I tried using the quadratic formula on the quadratic, and the discriminant was b^2-6b+1, which doesn't seem to help me in any way.
5) Prove [itex]a^4+b^4+c^4 >= a^2bc+b^2ac+c^2ab[/itex] for all real a,b,c.
Now usually when I encounter these type of questions It always happens to work that I minus the RHS from both sides, and the LHS can be factorized into some perfect square hence more or equal to zero. But I can't factorize it this time!
I know that if I get desperate I would use some multivariable calculus and find the global extrema of the function, But I would like to know how to do this with precalc methods thanks..6) Prove that if real numbers a, b and c satisfy [itex]a+b+c > 0, ab+ac+bc>0, abc > 0[/itex] then each a, b and c are positive.
I can't believe I can't do these...Sorry for the hassle guys, thanks!
1) Factor [itex]x^8+2x^4y^4+9y^8[/itex]
and [itex]a^4+b^4+c^2 - 2(a^2b^2 +a^2c+b^2c)[/itex]
I thought maybe it was some perfect square, the first one. I noticed the powers were quite coincidental so i let u=x^4 and v=y^4, but it seems there are no numbers that multiply to 9 and add to 2...I think that's irreducible. The second one I hoped I would notice its some expansion of something familiar, but I didnt get it..
Edit: I just tried Mathematicia and get [itex](x^4-2x^2y^2+3y^4)(x^4+2x^2y^2+3y^4) \mbox{and} (a^2-2ab+b^2-c)(a^2+2ab+b^2-c)[/itex] But I want to know how to get them manually!
2) Prove [itex]20^{22}-17^{22} +4^{33} -1[/itex] is divisible is 174.
I think i might need some mod arithmetic or something here, but I haven't done that before. Also, I noticed the differences, 20-17, 4-1, are the same, 3, which 174 is divisible by. And perhaps even more useless, that 174 is a 17 and 4 put together, which are in there...I tried expanding 20^22-17^22 and 4^33 -1 by [itex]x^n-1=(x-1)(1+x+x^2+x^3\cdots +x^{n-2}+x^{n-1})[/itex] and the similar one for x^n-y^n, but no go.
Edit: Once again mathematicia says the actualy number is 40768477197265827835864143774, and when divided by 174 gives 234301593087734642734851401, but I want to know how to do it.
3) Prove that if p,q,r and s are odd integers, then [itex]x^{10}+px^9-qx^7+rx^4-s=0[/itex] has no integer roots.
Just no idea on that one, none at all.
4) For which real values of b do the equations [itex]x^3+bx^2+2bx-1=0 \mbox{and} x^2+(b-1)x+b=0[/itex] have a common root?
I tried using the quadratic formula on the quadratic, and the discriminant was b^2-6b+1, which doesn't seem to help me in any way.
5) Prove [itex]a^4+b^4+c^4 >= a^2bc+b^2ac+c^2ab[/itex] for all real a,b,c.
Now usually when I encounter these type of questions It always happens to work that I minus the RHS from both sides, and the LHS can be factorized into some perfect square hence more or equal to zero. But I can't factorize it this time!
I know that if I get desperate I would use some multivariable calculus and find the global extrema of the function, But I would like to know how to do this with precalc methods thanks..6) Prove that if real numbers a, b and c satisfy [itex]a+b+c > 0, ab+ac+bc>0, abc > 0[/itex] then each a, b and c are positive.
I can't believe I can't do these...Sorry for the hassle guys, thanks!
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