MATH 103
Recommended problems from Review Exercises for Chapter 3, pages 178-181 (in Faires & DeFranza)

• Exercise 1 would be good except for the fact that none of the given graphs match any of the given polynomials!
• Do Exercise 2, but to make (c) and (d) consistent with the graphs that they are supposed to match, change their denominators to x+2 (from x+1).
• Do a couple from the group 3-6.
• Do a couple from the group 7-12.
• Do a couple from the group 13-16. Remember that a polynomial has as many zeros as its degree. But this includes any zeros that may be complex (as opposed to real) and it counts each zero a number of times equal to its multiplicity. Hence, the answer to (13) is three, because the zero x=1 has multiplicity two and the zero x=3 has multiplicity one. The sum of the multiplicities is three, so the polynomial's degree is at least three.

  Another rule you can use is that polynomial's degree is at least one more than the number of local extrema (minimums or maximums). In (13), for example, there is a local maximum at x=1 and a local minimum at x=2.2 (or thereabouts). As there are two local extrema, the degree of the polynomial must be at least three. In (16), there appear to be five local extrema (although it is not entirely clear), which would mean the polynomial's degree is at least six.

• Do a couple from the group 17-20. Use the polynomial division algorithm from page 142.
• Do a couple from the group 21-24. Here, you are given a polynomial and, in effect, one of its real factors, and you are to find the rest. (In the end, each factor should be either linear (e.g., x-3) or of higher degree but having no real zeros (e.g., x² + 3).
• Do a couple from the group 25-28. Recall that if a + bi is a zero of a polynomial, so is its complex conjugate, a - bi.

  Take (25), for example. We are given that 3+i is a zero of the polynomial, and so we know that 3-i is, as well. Which means that x-(3+i) and x-(3-i) are factors of the polynomial.

  To find the remaining zeros, begin by dividing the product (x-(3+i))(x-(3-i)) of the two factors we know, which comes out to x² - 6x + 10, into the given polynomial (using the polynomial division algorithm from page 142). The result is x² + x - 2, which is easily factored as (x+2)(x-1). As a final result, then, we get the product of all four factors:

  (x-(3+i))(x-(3-i))(x+2)(x-1)
• Do a couple from the group 29-34.
• Do a couple from the group 35-42.
• Do a few from the group 67-70. Be more concerned with finding an expression describing P than with its graph.
• Do at least one from among 71-72.