Question

Give 4 different reasons why the graph cannot be the graph of the polynomial $p$ given by
$p\left(x\right)=x^4-x^2+1$

Answer 1

1 - The given polynomial has degree $4$ and positive leading coefficient and the graph should rise on the left and right sides.

2 - $p\left(0\right)=1$, graph shows negative value.

3 - The equation $x^4-x^2+1=0$ has no solution which suggests that the polynomial $p\left(x\right)=x^4-x^2+1$ has no zeros. The graph shows x intercepts.

4 - The graph has a zero of multiplicity $2$, a zero of multiplicity $3$ and and a zero of multiplicity $1$. So the degree of the graphed polynomial should be at least $6$. The given polynomial has degree $4$.