Understanding Polynomial Functions and Graphs

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 • 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 Equations • Pg. 296-302 • 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 Equations • Pg. 312-317 • 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 Algebra • Pg. 319-324 • 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 • 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