Polynomials Observed

In perhaps the strangest element to my personality, I have eeked my way these last three months through a chapter in a math book on "Polynomial and Rational Functions." Here is my summary for review in perpetuity:

1. Quadratic Functions

• f(x)=ax2+bx+c
• standard form relating to parabola f(x)=a(x-h)2+k
• negative coefficient makes parabola go down; (h, k) is the vertex
• lead coefficient below 1 widens, greater narrows parabola
• x coordinate of vertex is -b/2a

2. Higher Degree Polynomials

Leading coefficient test: for even powered polynomials, positive is up on both sides; negative is down on both sides.

• Leading coefficient test: for odd powered polynomials, positive is down on left, up on right; negative is reversed.
• Turning points are at most n-1 times (where n is the degree of the polynomial)
• A function only has at most n real zeros
• Intermediate Value Theorem--everything in between two points

3. Synthetic Division

• This unit was pretty easy, basically long division with the coefficients of a polynomial.
• Remainder Theorem--divide the coefficients of a polynomial by a number and the remainder is the y point for the number as the x point.
• Factor Theorem--if the remainder is zero, then the number is a factor

4. Complex Numbers

• Also an easy section. Standard form is a +bi
• "Complex conjugates" are a+bi and a-bi. Multiplying them removes the imaginary part.

5. Zeros of Polynomial Functions

• Fundamental Theorem of Algebra--any polynomial has at least one zero :-)
• Linear Factorization Theorem--polynomials have as many factors as the degrees of n.
• Rational Zero Test--With integer coefficients, every rational zero is a factor of the final number divided by the lead coefficient.
• Complex zeros appear in conjugate pairs
• Descartes' Rule of Signs--# of positive real zeros is equal to the number of changes in sign in the polynomial or less than that by an even integer.
• Descartes continued--# of negative real zeros is equal to the number of changes in sign for f(-x) or less than that by an even integer.
• Upper Bound Rule: With a positive lead coefficient, c is an upper bound if, 1) it is positive and 2) the polynomial, divided by x-c, results in all positives in the bottom row of synthetic division (or all zeros), then c is an upper bound among the real zeros of the function.
• Lower Bound Rule: If 1) c is negative and 2) the numbers in the bottom row alternate between positive and negative (zero counts as an alternation) then c is a lower bound among the real zeros of the function.

6. Rational Functions

• In the form f(x)=N(x)/D(x)
• Domain obviously can't include any D(x) that equals 0 (vertical asymptote there).
• If the degree of the numerator is less than 0, then y=0 is a horizontal asymptote.
• If the degree of numerator and denominator are the same, then y=lead coefficient of numerator/lead coefficient of denominator is the horizontal asymptote.
• If the degree of numerator is more than denominator, no horizontal asymptote.
• If the degree of numerator is one more than denominator, then slant asymptote.  Divide the polynomial out with long division and the part before the remainder is the slant asymptote (in the form y=that bit)

7. Partial Fractions

• Decomposing rational functions into partial fractions: 1) divide if improper, 2) factor the denominator, 3) for each factor of the denominator (including repeated ones), you should have a rational component broken out, 4) quadratic factors are the hard ones--various (ax+c) in each numerator, the denominators build from the basic (ax2+bx+c) up the various powers of this e.g., (ax2+bx+c)2 and up to the power of the full quadratic.  All of course added together.
• The kind of repeatedness of #4 above applies to simple linear factors too.  So if (x+2)2 is a factor, then the broken out partial fractions will need to include both a denominator of (x+2) as well as one that is (x+2)2.
• John Bernoulli (1667-1748) developed the method of partial fractions.