Ch.3 Polynomial and Rational Functions Rachel Johnson Brittany Stephens

End Behavior A description of what happens as x becomes large in the positive or negative direction To describe, use the following notation x→ ∞ means “x becomes large in the positive direction” x → -∞ means “x becomes large in the negative direction”

Graphing Polynomial Functions 1.Zeros. Factor the polynomial to find all its real zeros; these are the x- intercepts of the graph. 2.Test Points. Make a table of values for the polynomial.

Graphing Polynomial Functions 3. End Behavior. Determine the end behavior of the polynomial. 4. Graph. Plot the intercepts and other points you found in the table. Sketch a smooth curve that passes through these points and exhibits the required end behavior.

Long Division Long division for polynomials is very much like long division for numbers.

Example

Solution

Synthetic Division A quick method of dividing polynomials Can be used when the divisor is in the form x-c Ex.

Example 3 | Solution:

Rational Zero Theorem p=factors of constant term q=factors of leading coefficient Every rational zero of P(x) is in the form

Finding the Rational Zeros of a Polynomial 1.List Possible Zeros. List all possible rational zeros using the Rational Zeros Theorem. 2.Divide. Use synthetic division to evaluate the polynomial at each of the candidates for rational zeros that found in Step 1. When the remainder is 0, note the quotient you have obtained.

Finding the Rational Zeros of a Polynomial 3. Repeat. Repeat Steps 1 and 2 for the quotient. Stop when you reach a quotient that is quadratic or factors easily, and use the quadratic formula or factor to find the remaining zeros.

Complex Zeros Once you have found all the rational zeros of an equation, you might have a quadratic equation left over from factoring. To find its zeros, use the quadratic formula Then, place the answer in a+bi form.

Sketching Graphs of Rational Functions 1.Factor. Factor the numerator and denominator. 2.Intercepts. Find the x-intercepts by determining the zeros of the numerator, and the y-intercept from the value of the function of x=0.

Sketching Graphs of Rational Functions 3. Vertical Asymptotes. Find the vertical asymptotes by determining the zeros of the denominator, and then see if y → ∞ or y → -∞ on each side of every vertical asymptote. 4. Horizontal Asymptote. Find the horizontal asymptote (if any) by dividing both numerator and denominator by the highest power of x that appears in the denominator, and then letting x → ∞

Sketching Graphs of Rational Functions 5. Sketch the Graph. Graph the information provided by the first four steps. Then plot as many additional points as needed to fill in the rest of the graph of the function.

Asymptotes 1.The vertical asymptotes of r are the lines x=a, where a is a zero of the denominator. 2.(a) If n<m, then r has horizontal asymptote y=0. (b) If n=m, then r has horizontal asymptote y= (c) If n>m, then r has no horizontal asymptote.