1. 4.5 Quadratic Equations Zero of the Function- a value where f(x) = 0 and the graph of the function intersects the x-axis Zero Product Property- for all numbers a and b, if ab = 0, then a = 0, b = 0, or both a = 0 and b = 0

2. 4.5 Quadratic Equations

4. 4.6 Completing the Square

5. 4.7 Quadratic Formula -

6. 4.8 Complex Numbers

8. 5.1 Polynomial Functions Monomial- a real number, a variable, or a product of a real number and one or more variables with whole number exponents Degree of a Monomial- in one variable is the exponent of the variable Polynomial- monomial or a sum of monomials Degree of a Polynomial- in one variable is the greatest degree among the its monomial terms

9. 5.1 Polynomial Functions -Standard Form of a Polynomial Function: 1.Coefficients (a) must be real #’s 2.Exponents must be positive integers 3.Domain = All Real #’s 4.Degree of a polynomial function is the highest degree of x (n)

10. 5.1 Polynomial Functions 1. Graphs of polynomials are smooth & continuous ; a turning point is where the graph changes directions 2. Leading Term Test for End Behavior: a) if n is odd and a n > 0  if n is odd and a n < 0  b) if n is even and a n > 0  if n is even and a n < 0  3. The graph can have at most n – 1 turning points

11. 5.2 Polynomials, Linear Factors, and Zeros - Real Zeros of Polynomial Functions: x = a is a zero of function f means  x = a is a solution of the equation f(x) = 0 means  (x – a) is a factor of f(x) means  (a,0) is an x-intercept of the graph of f -A function f can have at most n real zeros -Multiplicity of a zero—the # of times (x – a) occurs as a factor of f(x) “Even Multiplicity”  Graph touches the x-axis “Odd Multiplicity”  Graph crosses the x-axis

12. 5.2 Polynomials, Linear Factors, and Zeros -always measured on the x-axis -always named from Left to Right -always open brackets ( ) -Functions ONLY Local and Absolute Extrema: -local (relative) Maximum —the value of f(x) at the turning point when a graph goes from increasing to decreasing -local (relative) Minimum—the value of f(x) at the turning point when a graph goes from decreasing to increasing

13. 5.3 Solving Polynomial Equations Factored Polynomial- a polynomial is factored when it is expressed as a the product of monomials and polynomials Factoring by Grouping- when the terms and factors of a polynomial are grouped separately so that the remaining polynomial factors of each group are the same

14. 5.3 Solving Polynomial Equations Factoring by Grouping- Sum or Difference of Cubes-

15. 5.4 Dividing Polynomials -Synthetic Division: Given: ax 3 + bx 2 + cx + d divided by x – k Synthetic division method: 1.Add columns 2.Multiply by k a b c d k a kaka remainder

16. 5.4 Dividing Polynomials -Remainder Theorem: If a polynomial f(x) is divided by (x – k) then the remainder is r = f(k) -Factor Theorem: 1. If f(c) = 0, then (x – c) is a factor of f(x) 2. If (x – c) is a factor of f(x), then f(c) = 0

17. 5.5 Theorems About Roots of Polynomial Equations -Rational Zero Theorem: Every rational zero of f(x) has the form p/q, where: 1. p and q have no common factors other than 1 2. p is a factor of the constant term (a 0 ) 3. q is a factor of the leading coefficient (a n ) Complex Conjugate Theorem: if (a + bi) is a zero of f(x), then (a – bi) is also a zero

18. 5.6 The Fundamental Theorem of Algebra