Question #83abc

1 Answer
Oct 19, 2016

Answer:

No useful factorization for #f(x,y) = x^2 + 6xy + 9y^2 - 21y + 12#

Explanation:

#f(x,y) = x^2 + 6xy + 9y^2 - 21y + 12#

If #f(x,y)# is factorable, that implies in multiple tracing of #f(x,y)=0#

So, if #f(x,y)# is factorable for instance into #f_1(x,y)f_2(x,y)# this means that

#f(x,y)=f_1(x,y)f_2(x,y)=0# has a minimum of two null leafs. One for #f_1(x,y)=0# and the other for #f_2(x,y)=0#

In our case #f(x,y)# represents a slanted parabola having one null leaf, so this #f(x,y)# is not factorable.

We know that #f(x,y)# represents a parabola because putting it in the form

#f(x,y)=1/2(x,y)cdot M cdot((x),(y))+(x,y)cdot b + c#

#M=((2, 6),(6, 18))# has characteristic polynomial

#p_M(s) = s^2-20s=s(s-20)#

which has a null root.

Attached the null trace of #f(x,y)=0#

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