Warm Up Sept Rewrite using rational exponents: 2. Simplify: 3. Simplify: 4. Simplify: 5. Simplify:

September 3 rd, 2014 Operations with Polynomials

Polynomials A polynomial is a monomial or a sum of monomials (terms) Example: x 3 + xy + y 2

Types of Polynomials: Monomial – a #, variable or the product of the two. Binomial – the sum/difference of two monomials. example: x 2 + 2xy Trinomial – the sum/difference of three monomials. example: 5a – ab + c 4 Polynomials with more than three terms do not have a special name. Classifying Polynomials

Polynomial Standard Form  Degree of a monomial – the sum of the exponents of all its variables.  Degree of a polynomial – the greatest degree of any one term in the polynomial. (you must find the degree of each term).  Standard Form of Polynomials- is ordering terms in descending (decreasing) order by degree.

Example 1: Arrange the terms of each polynomial so that the polynomial is in Standard Form. 1. 6x – 8x – 2x x 2 + 2x 4 – 11

Add & Subtract Polynomials Adding Polynomials Hint: combine like terms! Make sure they have the same variable & exponent. (4xy + 7x 2 – 6y) + (3xy + 2y – 5x 2 ) We are NOT multiplying or dividing so… DON’T TOUCH THE EXPONENTS!!!!

Example 1: Find the sum of (4x 2 + 3x - 7) + (x )

Watch out for mixed up polynomials! Example 2: Find the sum of (7 + 3x 2 + 5xy) + (xy – 2x 2 + 2)

Subtracting Polynomials Hint: distribute the subtraction sign then combine like terms! Example 3 : Find the difference. (3x 2 + 2x – 6) – (2x + x 2 + 3)

Example 4: Find the difference. (5ab 2 + 3ab) – (2ab – 8ab)

Geometry Application Example 5: Given the perimeter and the measures of 2 sides of a triangle, find the measure of the third side. P = 7x + 3y x – 2y 2x + 3y

Review How would you multiply 3(5x – 1) ? Can we classify these polynomials?

Multiplying a MONOMIAL and a POLYNOMIAL Two things to remember: 1. Use the DISTRIBUTIVE PROPERTY ! 2. When multiplying variables, ADD the exponents. Example 1 :

Examples 2 & 3:

Example 4: What is different here?

Example 5:

Example 6: You want to find the area of the classroom. Your teacher tells you that the length is 5 feet less than twice the width. Write a single polynomial to express the area of the room.

Can we classify these 2 polynomials? Example 9: (2x + 3)(5x + 8)

Multiplying a BINOMIAL and a BINOMIAL We STILL use the DISTRIBUTIVE PROPERTY. But we also have some special tricks to make distributing easier: FOIL Box Method

FOIL FOIL is an acronym that can help you multiply two binomials. F – First (2x + 3)(5x + 8) O – Outside I – Inside L – Last

Box Method Draw a box and write one binomial on the top and the other on the bottom. Multiply each pair of terms. Your answer is on the inside of the box. Combine like terms to write your final answer. Example 7: (3x – 5)(5x + 2)

Let’s see how it works… Example 8) Example 9) (8x + 1)(x – 3) (2x + 3)(5x + 8)

A Binomial SQUARED What does it mean to SQUARE a number? How could we simplify the expression Example 15) (4x + 1) 2

Example 10 (2x – 3y) 2

Can we classify the polynomials below? Example 11) (3x + 7)(2x 2 – x + 5)

Examples 12 & 13: (r – 2)(3r 2 + 4r – 1) (2a + 3)(a + b)(2a – 7)

What about this? You try! Example 14 (2x - 5) 3 =

Example 14: Find the area of the rectangle below:

Challenge Questions for candy! Simplify: -4b(2b + 1) – 8(b 2 + 2b – 2) Simplify: x 2 (x + 1) + 5x(x – 3) – 4(x + 10)

Homework Work on Weekly HW #2 Work needs to be shown on a separate sheet of NB paper with final answers highlighted or circled.