Question 1177786
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The response from the other tutor has nothing to do with the question that is asked.<br>
Her response is about PAIRS of equations which have no COMMON SOLUTION.  The question is about single polynomial equations that have no REAL ROOTS.<br>
A polynomial equation of odd degree will always have at least one real solution, because the end behavior for large positive x is different than for large negative x.<br>
A polynomial equation of even degree does not need to have a real root, because the end behavior for large positive x and for large negative x is the same.  If a polynomial equation of even degree has all coefficients positive, then every term will be positive and the equation will have no real solutions; and likewise if all terms have negative coefficients.<br>
Here are graphs of two polynomial equations with no real solutions:
{{{y=x^2+3}}}
{{{y=-x^4-2x^2-1}}}<br>
{{{graph(400,400,-3,3,-40,40,x^2+3,-x^4-2x^2-1)}}}<br>