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 153012 by bobhans last updated on 04/Sep/21

 Find all ordered pairs of real    numbers (x,y) for which     { (((1+x^4 )(1+x^2 )(1+x)=1+y^7 )),(((1+y^4 )(1+y^2 )(1+y)=1+x^7 )) :}

$$\:{Find}\:{all}\:{ordered}\:{pairs}\:{of}\:{real}\: \\ $$$$\:{numbers}\:\left({x},{y}\right)\:{for}\:{which} \\ $$$$\:\:\begin{cases}{\left(\mathrm{1}+{x}^{\mathrm{4}} \right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}\right)=\mathrm{1}+{y}^{\mathrm{7}} }\\{\left(\mathrm{1}+{y}^{\mathrm{4}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}\right)=\mathrm{1}+{x}^{\mathrm{7}} }\end{cases} \\ $$

Commented by mr W last updated on 04/Sep/21

f(x)=f^(−1) (x)  ⇒y=x  (1+x^4 )(1+x^2 )(1+x)=1+x^7   x(1+x+x^2 +x^3 +x^4 +x^5 )=0  ((x(1−x^6 ))/(1−x))=0  ⇒x=0, −1  ⇒solution is (0,0) or (−1,−1)

$${f}\left({x}\right)={f}^{−\mathrm{1}} \left({x}\right) \\ $$$$\Rightarrow{y}={x} \\ $$$$\left(\mathrm{1}+{x}^{\mathrm{4}} \right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}\right)=\mathrm{1}+{x}^{\mathrm{7}} \\ $$$${x}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +{x}^{\mathrm{4}} +{x}^{\mathrm{5}} \right)=\mathrm{0} \\ $$$$\frac{{x}\left(\mathrm{1}−{x}^{\mathrm{6}} \right)}{\mathrm{1}−{x}}=\mathrm{0} \\ $$$$\Rightarrow{x}=\mathrm{0},\:−\mathrm{1} \\ $$$$\Rightarrow{solution}\:{is}\:\left(\mathrm{0},\mathrm{0}\right)\:{or}\:\left(−\mathrm{1},−\mathrm{1}\right) \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com