Polynomials and Polynomial Functions Chapter 5

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

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

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

5.1 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

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

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.

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

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

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

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

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)

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

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

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.

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.

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

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