Topics:
• More on complete graphs of polynomials and recognizing when we do not have a complete graph.
Facts about polynomial graphs:
• The domain of a polynomial is the whole real line
• The y-intercept is the constant term
• For a polynomial of degree n, there are at most n real roots (which are the same as the x-intercepts of its graph) and there are at most n-1 local maxima or minima (turning points).
• The graph of a polynomial is continuous – no breaks or jumps
• The graph of a polynomial does not have any corners
• The graph of a non-constant polynomial does not contain any horizontal line segment
• The end behavior of the graph is determined by the leading term of the polynomial.
• Review of complex numbers
• The fundamental theorem of algebra and its consequences.
The important thing to remember is that the Fundamental Theorem implies that every polynomial of degree n>0 has exactly n roots, if you allow complex numbers as roots and you count each root with its multiplicity. (The multiplicity of a root is the number of times its factor appears in the factorization of f(x), in other words, it is the exponent that goes on that factor.)
• Finding all roots of a polynomial and using them to find a complete factorization of the polynomial over the complex numbers.
• For a polynomial with real coefficients, if it has non-real complex roots they occur in conjugate pairs: if a+bi is a root, then so is a-bi.
Homework:
• Review the examples discussed in class, and also study Example 10.9 in the textbook.
• Optional, but recommended: Do Exercise 9.4(a-d). In each case, start with the standard viewing window. Write down each change you make to the standard viewing window and why you made it – what was it about the graph that you wanted to see that you could not see.
• Do the assigned parts from Session 10 Exercises 10.3 and 10.4
• Do the WeBWorK – due by Monday 11 PM
• Do the Warm-Up for Rational Functions – also due by Monday 11 PM
• Don’t forget that Test 2 is scheduled for Thursday 20 March. There will be a separate post with more information.
