Chapter 6 – Polynomial Functions Algebra 2

Warm Up

6-1 Polynomial Functions A monomial is an expression that is either a real number, a variable or a product of real numbers and variables. A polynomial is a monomial or the sum of monomials. The exponent of the variable in a term determines the degree of that term. Standard form of a polynomial has the variable in descending order by degree.

6-1 Polynomial Functions

6-1 Polynomial Functions The degree of a polynomial is the greatest degree of any term in the polynomial

6-1 Polynomial Functions Write each polynomial in standard form and classify it by degree.

6-2 Polynomials and Linear Factors You can write a polynomial as a product of its linear factors

6-2 Polynomials and Linear Factors You can sometimes use the GCF to help factor a polynomial. The GCF will contain variables common to all terms, as well as numbers

6-2 Polynomials and Linear Factors

6-2 Polynomials and Linear Factors

6-2 Polynomials and Linear Factors If a linear factor of a polynomial is repeated, the zero is repeated. A repeated zero is called a multiple zero. A multiple zero has multiplicity equal to the number of times the zero occurs.

6-2 Polynomials and Linear Factors

6-2 Polynomials and Linear Factors page 323 (1-11, 17-35)odd you do NOT need to graph the functions.

warm up

6-3 Dividing Polynomials

Polynomial Long Division Two people per worksheet. Take turns at each step, first partner decides what you multiply the divisor by, second partner agrees and does the multiplication, first partner agrees and does the subtraction, then switch for next term. You may do the work on the worksheet, paper or the white board. If you use the white board you must have me check EACH answer as you complete it.

Synthetic Divison Warm Up: Write a polynomial function in standard form with zeros at -1, 2 and 5. Use long division to divide: Use long division to divide x3 – 6x2 + 3x + 10 x3 – x2 +1 x3 – 2x2 –x + 6

Synthetic Division

6-4 Solving polynomial equations

Solve for all three roots

solving using a quadratic model

Homework: page 330 (227-33) odd page 336 (13 – 31) odd,

warm up Solve these equations: 1. x3 + 125 = 0 2. x4 + 3x2 – 28 = 0

6-5 Theorems about roots

6-5 Theorems about roots

6-5 Theorems about roots To find all the roots of a polynomial: determine the possible rational roots using the rational root theorem (ao/an) Use synthetic division to test the possible rational roots until one divides evenly Write the factored form and solve for all roots Use the quadratic formula if necessary You may need to use synthetic division more than once

6-5 Theorems about roots

6-5 Theorems about roots Warm Up Find the polynomial equation in standard form that has roots at -5, -4 and 3 Find f(-2) for f(x) = x4 – 2x3 +4x2 + x + 1 using synthetic division Solve x4 – 100 = 0

6-5 Theorems about roots Practice Problem: List all the possible rational roots of 3x3 + x2 – 15x – 5 = 0 Use synthetic division to determine which of these is a root Factor and solve for the rest of the roots of the equation.

6-5 Theorems about roots

6-5 Theorems about roots A third degree polynomial has roots 2 and √3. Write the polynomial in standard form.

6-5 Theorems about roots

6-5 Theorems about roots

6-5 Theorems about roots

6-5 Theorems about roots

6-5 Theorems about roots Homework p 345 (11-23) odd

6-6 Fundamental Theorem of Algebra

6-6 Fundamental Theorem of Algebra

6-6 Fundamental Theorem of Algebra

6-7 Permutations and Combinations

A selection of items in which order does not matter is called a combination

homework p 354 (1-29) odd

6-8 The Binomial Theorem Warm Up Find the zeros of the function by finding the possible rational roots and using synthetic division. multiply each and write in standard form: (x + y)2 (x + y)3 (x + y)4

6-8 The Binomial Theorem Notice that each set of coefficients matches a row of Pascal’s Triangle Each row of Pascal’s Triangle contains coefficients for the expansion of (a+b)n For example, when n = 6 you can find the coefficients for the expansion of (a+b)6 in the 7th row of the triangle. Use Pascal’s Triangle to expand (a+b)6

6-8 The Binomial Theorem If the terms of the polynomial have coefficients other than 1, you can still base the expansion on the triangle.

6-8 The Binomial Theorem Evaluate 4C0 4C1 4C2 4C3 4C4

6-8 The Binomial Theorem

6-8 The Binomial Theorem Warm up

6-8 The Binomial Theorem To find a particular term of a binomial expansion you do not need to calculate the entire polynomial! Ex: Find the 5th term of (x – 4)8 Find the 4th term of (x – 3)8

6-8 The Binomial Theorem Chapter 6 Test this Thursday (5th) or Friday (4th) Homework: Complete practice test