Sunday, 29 January 2017

polynomials - How do you find roots of an equation which is a sum of quadratic equations?



So the questions says -



Let f(x),g(x) and h(x) be quadratic polynomials having positive leading coefficients and real and distinct roots. If each pair of them has a common root, then find roots of f(x)+g(x)+h(x)=0.




What I did -



Let,



f(x)=a1(xα)(xβ),g(x)=a2(xβ)(xγ),h(x)=a3(xγ)(xα),F(x):=f(x)+g(x)+h(x)


Now,



F(α)=a2(αβ)(αγ)F(β)=a3(βγ)(βα)F(γ)=a1(γα)(γβ)




I don't know how to proceed further. I referred to the solution, it just multiplies F(α),F(β), and F(γ) and it comes out to be negative. And hence it concludes that roots of F(x)=0 are real and distinct. Can anyone explain why?



Thanks.


Answer



Suppose, without loss of generality, that α<β<γ: it is easy to check that F(α)>0, F(β)<0 and F(γ)>0. But F(x) is a quadratic polynomial, hence a continuous function: it follows that F(x)=0 for some x between α and β, and also for some x between β and γ.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...