 # Algebra 2.  Warm Up  A monomial is an expression that is either a real number, a variable or a product of real numbers and variables.  A polynomial.

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Algebra 2

 Warm Up

 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.

 The degree of a polynomial is the greatest degree of any term in the polynomial

 Write each polynomial in standard form and classify it by degree.

You can write a polynomial as a product of its 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

 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.

 page 323 (17-35)odd  you do NOT need to graph the functions.

 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.

 Warm Up: 1. Write a polynomial function in standard form with zeros at -1, 2 and 5. 2. Use long division to divide: 3. Use long division to divide

Steps: Identify cubes Rewrite in cubed form Use formulas to factor

 Solve for all three roots

 Factor each polynomial completely

 Homework:  page 336 (13 – 31) odd,

 Solve these equations:  1. x 3 + 125 = 0  2. x 4 + 3x 2 – 28 = 0

 Warm Up  Find the polynomial equation in standard form that has roots at -5, -4 and 3  Find f(-2) for f(x) = x 4 – 2x 3 +4x 2 + x + 1 using synthetic division

 To find all the roots of a polynomial: ◦ determine the possible rational roots using the rational root theorem (a o /a n ) ◦ 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

 Practice Problem:  List all the possible rational roots of ◦ 3x 3 + x 2 – 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.

 A third degree polynomial has roots 2 and √3. Write the polynomial in standard form.

 Homework  p 345 (11-23) odd

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

 homework  p 354 (1-29) odd

 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

 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 7 th row of the triangle.  Use Pascal’s Triangle to expand (a+b) 6

 If the terms of the polynomial have coefficients other than 1, you can still base the expansion on the triangle.

 Evaluate 4 C 0 4 C 1 4 C 2 4 C 3 4 C 4

Warm up

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

 Chapter 6 Test this Thursday (5 th ) or Friday (4 th )  Homework: Complete practice test

 Read the rules for exponents on page 374 and copy them into your notes.  Complete all problems (1-25) on page 374

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