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 \(f(b).\)

E 2.2 Exercises