APC Unit 3 CH-4.5 Real Zeros, Long And synthetic division Remainder theorem, Rational Zero Test

Warm-up  Find the value(s) of (k) when (x+1) is a factor of k 2 x 4 – 2kx 2 –x – 9 Do the problem Review the answers with your desk groups Did anyone do it a better way?

Warm-up Key Concepts  What does it mean to be a factor  Divides evenly  Remainder is zero  Synthetic Division  Remainder theorem  Just plug in (-1)  In this case set the expression = 0.

Unit 3  Polynomials  What is/What is not a polynomial  Characteristics  Zeros (even – touches) (odd – Crosses)  End Behavior  Lead coefficient  degree  Intercepts  Maximums/Minimums  (with calculator unless it is a quadratic)  Increasing/decreasing

Sketch a polynomial function  5th degree  With a lead coefficient of (-2)

Graphing a polynomial  (be sure to read the question carefully and include all the requirements)

Given a graph – find the function  How  Look at zeros -  Change to factors  Look at multiplicity of each zero  Raise the factor to that power  Look at end behavior to determine if lead coefficient, a, is positive or negative  Look at y-intercept or other given point to solve for a

Rational Functions  What is a rational Function  Characteristics  Domain  Vertical Asymptotes  Holes  Horizontal asymptotes  Zeros  Intercepts  Maximums/Minimums  (with calculator)  Increasing/decreasing

Rational Function – Keep clear  Domain (all real values except vertical asymptotes and holes)  Where the denominator = 0  (don’t simplify – we want the domain of the given function not an equivalent function)  Vertical asymptote  Where the denominator = 0 and the numerator not equal to zero  Holes  Where the denominator = 0 and the numerator =0  How to find…  Factor both the numerator and denominator  X value that make the common factor = 0  Cancel common factors and plug x value into the equivalent expression  y value of the hole

Horizontal Asymptotes  D > N y=0  D = N y=a/b  D < N none  ** D = N-1 Oblique asymptote use long division

Graphing Rational functions Vertical asymptotes use dotted lines and label Horizontal/oblique asymptotes use dotted lines and label Plot zeros, y-intercepts (if any) Place holes in the right spot (o) Pick points left and right of VA Draw smooth curve from point towards asymptotes

Given the graph find the rational function  Change zeros to factor -  numerator  Change asymptotes to factors  Denominator  Change hole to factor  both N and D  Use intercept to solve for a  Check end behavior

How to approach solving higher order polynomials  Start with listing possible zeros p/q  Try x =1 use remainder theorem and end behavior to decide next value to try  Skip a value (or 2) use remainder theorem if one value has a positive remainder and the next value has a negative remainder the zero is in-between  When a zero is found, then use the reduced expression until a 2 nd degree is found, then factor or use quadratic formula

Higher Order Polynomials  If it is on the calculator section  Graph and find the zeros with your calculator

Complex Zeros  Always come in Complex Conjugates  If a + bi is a zero  Then we also know that…  a – bi is also a zero

Inequalities  Set one side = 0  Solve for key values  Zeros  VA  Use number line and check points in between  Evaluate each key value