Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 173798 by payamnazerian last updated on 18/Jul/22

$$$$$${({\phi }^5+{\phi }^4+1)}^2=m\phi +n$$$$m,n=?$$$$........$$$$$$

Answered by mr W last updated on 18/Jul/22

$${\phi }^2-\phi -1=0$$$${\phi }^2=\phi +1$$$${\phi }^3={\phi }^2+\phi =2\phi +1$$$${\phi }^4=2{\phi }^2+\phi =3\phi +2$$$${\phi }^5=3{\phi }^2+2\phi =5\phi +3$$$${\phi }^5+{\phi }^4+1=5\phi +3+3\phi +2+1=8\phi +6$$$${({\phi }^5+{\phi }^4+1)}^2=64{\phi }^2+96\phi +36$$$${({\phi }^5+{\phi }^4+1)}^2=64(\phi +1)+96\phi +36$$$${({\phi }^5+{\phi }^4+1)}^2=160\phi +100=m\phi +n$$$$\Rightarrow m=160,n=100$$

Commented by Tawa11 last updated on 18/Jul/22

$$\mathrm{Great}\mathrm{sir}$$

Commented by mnjuly1970 last updated on 18/Jul/22

$$gratefulsir$$

Answered by Rasheed.Sindhi last updated on 18/Jul/22

$$Anotherway$$$${({\phi }^5+{\phi }^4+1)}^2=m\phi +n;m,n=?$$$$$$$${\phi }^2-\phi -1=0\Rightarrow {\phi }^2=\phi +1\Rightarrow {\phi }^2-1=\phi$$$$$$$${\phi }^5+{\phi }^4+1={\phi }^4(\phi +1)+1$$$$={\phi }^4{\phi }^2+1={\phi }^6+1$$$$=({\phi }^2+1)({\phi }^4-{\phi }^2+1)$$$$=({\phi }^2-1+2)({\phi }^2({\phi }^2-1)+1)$$$$=(\phi +2)\{\phi (\phi +1)+1\}$$$$=(\phi +2)({\phi }^2+\phi +1)$$$$=(\phi +2)({\phi }^2-\phi -1+2\phi +2)$$$$=(\phi +2)(0+2\phi +2)$$$$=2{\phi }^2+6\phi +4$$$$=2(\phi +1)+6\phi +4$$$$=8\phi +6$$$${({\phi }^5+{\phi }^4+1)}^2={(8\phi +6)}^2$$$$=64{\phi }^2+96\phi +36$$$$=64(\phi +1)+96\phi +36$$$$=160\phi +100$$$$m=160,n=100$$

Commented by mnjuly1970 last updated on 19/Jul/22

$$thanksalotsir$$

Answered by Rasheed.Sindhi last updated on 18/Jul/22

$$Stillanotherway$$$${\phi }^2-\phi -1=0$$$${\phi }^2=\phi +1$$$${\phi }^2+\phi =2\phi +1$$$${\phi }^5+{\phi }^4={\phi }^3(2\phi +1)$$$$={\phi }^2(2{\phi }^2+\phi )$$$$=(\phi +1)(2(\phi +1)+\phi )$$$$=(\phi +1)(3\phi +2)$$$$=3{\phi }^2+5\phi +2$$$$=3(\phi +1)+5\phi +2$$$${\phi }^5+{\phi }^4+1=8\phi +5+1=8\phi +6$$$${({\phi }^5+{\phi }^4+1)}^2={(8\phi +6)}^2$$$$=64{\phi }^2+96\phi +36$$$$=64(\phi +1)+96\phi +36$$$$=160\phi +100$$$$m=160,n=100$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com