T the undertaker May 2009 29 0 May 8, 2010 #1 Prove that if p,q,r and s are odd integers, then this equation has no integer roots: x^10 + px^9 - qx^7 + rx^4 - s = 0 Last edited by a moderator: May 11, 2010

Prove that if p,q,r and s are odd integers, then this equation has no integer roots: x^10 + px^9 - qx^7 + rx^4 - s = 0

S Soroban MHF Hall of Honor May 2006 12,028 6,341 Lexington, MA (USA) May 8, 2010 #2 Hello, the undertaker! I baby-talked my way through it . . . Prove that if \(\displaystyle p,q,r, s\) are odd integers, then: .\(\displaystyle x^{10} + px^9 - qx^7 + rx^4 - s \:=\: 0\) .has no integer roots. Click to expand... Note that: An even integer raised to any positive integral power is even: .\(\displaystyle \text{(even)}^n \:=\:\text{even}\) An odd integer raised to any positive integral power is odd: .\(\displaystyle \text{(odd)}^n \:=\:\text{odd}\) We have: .\(\displaystyle \bigg[x^{10} + px^9 + rx^4\bigg] - \bigg[qx^7 + s\bigg] \;=\;0\) Suppose \(\displaystyle x\) is even. We have: .\(\displaystyle \bigg[(even)^{10} + p(even)^9 + r(even)^4\bigg] - \bigg[q(even)^7 + (odd)\bigg] \;=\;0 \) . . . . . . . . . \(\displaystyle \underbrace{\bigg[(even) + (even) + (even)\bigg]}_{\text{(even)}} - \underbrace{\bigg[(even) + (odd)\bigg]}_{\text{(odd)}} \;=\;0 \) And the difference of an even number and an odd number cannot be zero. Suppose \(\displaystyle x\) is odd. We have: .\(\displaystyle \bigg[(odd)^{10} + p(odd)^9 + r(odd)^4\bigg] - \bigg[r(odd)^4 + (odd)\bigg] \;=\;0 \) . . . . . . . . . \(\displaystyle \underbrace{\bigg[(odd) + (odd) + (odd)\bigg]}_{\text{(odd)}} - \underbrace{\bigg[(odd) + (odd)\bigg]}_{\text{(even)}} \;=\;0\) And the difference of an odd number and an even number cannot be zero. Reactions: Bacterius

Hello, the undertaker! I baby-talked my way through it . . . Prove that if \(\displaystyle p,q,r, s\) are odd integers, then: .\(\displaystyle x^{10} + px^9 - qx^7 + rx^4 - s \:=\: 0\) .has no integer roots. Click to expand... Note that: An even integer raised to any positive integral power is even: .\(\displaystyle \text{(even)}^n \:=\:\text{even}\) An odd integer raised to any positive integral power is odd: .\(\displaystyle \text{(odd)}^n \:=\:\text{odd}\) We have: .\(\displaystyle \bigg[x^{10} + px^9 + rx^4\bigg] - \bigg[qx^7 + s\bigg] \;=\;0\) Suppose \(\displaystyle x\) is even. We have: .\(\displaystyle \bigg[(even)^{10} + p(even)^9 + r(even)^4\bigg] - \bigg[q(even)^7 + (odd)\bigg] \;=\;0 \) . . . . . . . . . \(\displaystyle \underbrace{\bigg[(even) + (even) + (even)\bigg]}_{\text{(even)}} - \underbrace{\bigg[(even) + (odd)\bigg]}_{\text{(odd)}} \;=\;0 \) And the difference of an even number and an odd number cannot be zero. Suppose \(\displaystyle x\) is odd. We have: .\(\displaystyle \bigg[(odd)^{10} + p(odd)^9 + r(odd)^4\bigg] - \bigg[r(odd)^4 + (odd)\bigg] \;=\;0 \) . . . . . . . . . \(\displaystyle \underbrace{\bigg[(odd) + (odd) + (odd)\bigg]}_{\text{(odd)}} - \underbrace{\bigg[(odd) + (odd)\bigg]}_{\text{(even)}} \;=\;0\) And the difference of an odd number and an even number cannot be zero.

Bacterius Nov 2009 927 260 Wellington May 8, 2010 #3 Geez, that was awesome Soroban. Such a simple technique for such a seemingly intractable problem ! Too bad I can't triple-thank you for this one. The only improvement I can suggest is putting the "even" and "odd" tags between \mathrm{}, like \(\displaystyle \mathrm{even}\). Otherwise, (Clapping)

Geez, that was awesome Soroban. Such a simple technique for such a seemingly intractable problem ! Too bad I can't triple-thank you for this one. The only improvement I can suggest is putting the "even" and "odd" tags between \mathrm{}, like \(\displaystyle \mathrm{even}\). Otherwise, (Clapping)