1. Polynomials and Polynomial Functions
Chapter 5

2. 5.1 Polynomial Functions Pg. 280-287
Obj: Learn how to classify and graph polynomials and describe end behavior.
F.IF.7.c, A.SSE.1.a

3. 5.1 Polynomial Functions
Monomial – a real number, a variable, or a product of a real number and one or more variables
Degree of a Monomial – the exponent of the variable
Polynomial – a monomial or a sum of monomials
Degree of a Polynomial – the greatest degree among its monomial terms
Polynomial Function – a polynomial with the variable x
Standard Form of a Polynomial Function – arranges the terms by degree in descending numerical order

4. 5.1 Polynomial Functions Naming Polynomials Naming Polynomials Degree
1- linear
2 – quadratic
3 – cubic
4 – quartic
5 - quintic
Naming Polynomials
Number of Terms
1 – Monomial
2 – Binomial
3 – Trinomial
4 – Polynomial of 4 Terms

5. 5.1 Polynomial Functions
Turning Point – the degree of a polynomial affects its shape and the number of turning points
Degree n has at most n-1 turning points
End Behavior - The directions of the graph to the far left and to the far right
Increasing – when the y-values increase as the x-values increase
Decreasing – when the y-values decrease as the x-values increase
End Behavior of a Polynomial Function with Leading term axⁿ
a positive
n even – Up and Up
n odd – Down and Up
a negative
n even – Down and down
n odd – Up and down

6. 5.2 Polynomials, Linear Factors, and Zeros
Pg. 288 – 295
Obj: Learn how to analyze the factored form of a polynomial and write a polynomial function from its zeros.
F.IF.7.c, A.APR.3

7. 5.2 Polynomials, Linear Factors, and Zeros
Factor Theorem – x-a is a factor of a polynomial if and only if the value of a is a zero of the related polynomial function
Multiple Zero – a zero that occurs more than once
Multiplicity – “a is a zero of multiplicity n” means that x-a appears n times as a factor
How multiple zeros affect a graph – If a is a zero of multiplicity n in the polynomial function y=P(x), then the behavior of the graph at the x- intercept a will be close to linear if n=1, close to quadratic if n=2, close to cubic if n=3, and so on.

8. 5.2 Polynomials, Linear Factors, and Zeros
Relative Maximum – the value of the function at an up-to-down turning point
Relative Minimum – the value of the function at a down-to-up turning point

9. 5.3 Solving Polynomial EquationsPg
Obj: Learn how to solve polynomial equations by factoring and graphing.
A.REI.11, A.SSE.2

10. 5.3 Solving Polynomial Equations
Polynomial Factoring Techniques
Factoring out the GCF
Quadratic Trinomials
Perfect Square Trinomials
Difference of Squares
Factoring by Grouping
Sum or Difference of Cubes

11. 5.4 Dividing Polynomials Pg. 303-310
Obj: Learn how to divide polynomials using long division and synthetic division.
A.APR.2, A.APR.1, A.APR.6

12. 5.4 Dividing Polynomials
Synthetic Division – Simplifies the process of long-division – write the coefficients (including zeros) of the polynomial in standard form – omit all variables and exponents – for the divisor reverse the sign (this allows you to add instead of subtract throughout the process)
Remainder Theorem – If you divide a polynomial P(x) of degree n>1 by x- a, then the remainder is P(a)

13. 5.5 Theorems about Roots of Polynomial EquationsPg
Obj: Learn how to solve equations using the Rational Root Theorem and use the Conjugate Root Theorem.
N.CN.7, N.CN.8

14. 5.5 Theorems about Roots of Polynomial Equations
Rational Root Theorem
Integer roots must be factors of aₒ
Rational roots must have reduced form p/q where p is an integer factor of aₒ and q is an integer factor of a

15. 5.5 Theorems about Roots of Polynomial Equations
Conjugate Root Theorem
If P(x) is a polynomial with rational coefficients, then irrational roots of P(x)=0 that have the form a+b occur in conjugate pairs. That is if a+b is an irrational root with a and b rational, then a-b is also a root.
If P(x) is a polynomial with real coefficients, then the complex roots of P(x)=0 occur in conjugate pairs. That is, a+bi is a complex root with a and b real, then a-bi is also a root.

16. 5.5 Theorems about Roots of Polynomial Equations
Descartes’ Rule of Signs
Let P(x) be a polynomial with real coefficients written in standard form
The number of positive real roots of P(x) = 0 is either equal to the number of sign changes between consecutive coefficients of P(x) or is less than that by an even number.
The number of negative real roots of P(x) = 0 is either equal to the number of sign changes between consecutive coefficients of P(-x) or is less than that by an even number.
In both cases count multiple roots according to their multiplicity.

17. 5.6 The Fundamental Theorem of AlgebraPg
Obj: Learn how to use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions.
N.CN.7, N.CN.8, N.CN.9, A.APR.3

18. 5.6 The Fundamental Theorem of Algebra
If P(x) is a polynomial of degree n>1, then P(x) = 0 has exactly n roots, including multiple and complex roots.
Equivalent ways to state the Fundamental Theorem of Algebra
Every polynomial equation of degree n > 1 has exactly n roots, including multiple and complex roots
Every polynomial of degree n > 1 has n linear factors
Every polynomial function of degree n > 1 has at least one complex zero