Determine whether the following statement is true: if the graph of a polynomial does not cross the 𝑥-axis, then the degree of the polynomial is even.
A polynomial is either even or it’s odd. If it’s even with a positive leading coefficient, both arrows will go up. If it’s an even polynomial with a negative leading coefficient, both arrows will go down.
Now what happens in between the arrows can vary. If it’s an odd polynomial with a positive leading coefficient, it’ll increase left to right — the arrows go in opposite directions — and then opposite for a negative leading coefficient for an odd-degree polynomial.
So in order for a graph of a polynomial to not cross the 𝑥-axis, the arrows can never go in that direction. So it may be something like this. This means it would be an even-degree polynomial, so this would be a true statement.