SOLUTION: Explain why the polynomial function defined by f(x)=x^4+6x^2+2 cannot have any real zeros

Question 412903: Explain why the polynomial function defined by f(x)=x^4+6x^2+2 cannot have any real zeros
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
Explain why the polynomial function defined by f(x)=x^4+6x^2+2 cannot have any real zeros
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The reason this function cannot have any real zeros is that this is a parabola that opens upward with a minimum f(x)=+2. That is, the range is [2,infinity). The graph below clearly shows this. This can also be shown algebraically as follows:
let x^2=u
then x^4=u^2
u^2+6u+2=0
using quadratic formula to solve,
a=1, b=6,c=2
u=(-6+-sqrt(b^2-4*1*2))/2
u=(-6+-sqrt(28))/2=-6+-(5.29)/2
u=-11.29 or -.71
note that two of the u roots are negative,so x can not have any real solutions
since x^2=u.
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