# 2.2 | Polynomial Functions of Higher Degree

The Leading Term Test The end-behavior of a polynomial function is determined by its leading term. That is,
$$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots +a_1 x + a_0,$$
has the same end-behavior as
$$g(x) = a_n x^n.$$
Moreover,

• if $$n$$ is even then $$g(x) = a_n x^n$$ has the same end-behavior as $$h(x) = a_n x^2$$; and
• if $$n$$ is odd then $$g(x) = a_n x^n$$ has the same end-behavior as $$h(x) = a_n x$$.
For any polynomial
$$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots +a_1 x +a_0,$$
then $$f$$ has the same end-behavior as

• $$g(x) = x^2,$$ if $$a_n > 0$$ and $$n$$ is even.
• $$g(x) = – x^2,$$ if $$a_n < 0$$ and $$n$$ is even.
• $$g(x) = x,$$ if $$a_n >0$$ and $$n$$ is odd.
• $$g(x) = -x,$$ if $$a_n <0.$$ and $$n$$ is odd.
A factor $$(x-a)^k,$$ $$k>1,$$ yields a repeated zero $$x = a$$ of multiplicity $$k.$$

1. If $$k$$ is odd, the graph crosses the $$x$$-axis at $$x = a$$.
2. If $$k$$ is even, the graph touches the $$x$$-axis (but does not cross the $$x$$-axis) at $$x = a$$.
Intermediate Value Theorem Let $$a$$ and $$b$$ be real numbers such that $$a < b$$. If $$f$$ is a polynomial function such that $$f(a) \not = f(b),$$ then, in the interval $$[a,b],$$ $$f$$ takes on every value between $$f(a)$$ and $$f(b).$$