Warm Up Jan. 28th 1. Rewrite using rational exponents: 2. Rewrite in radical form then simplify the radical: 3. Simplify: 4. Simplify: 5. Simplify: 6. Simplify:

Operations with Polynomials

Polynomials A polynomial is a monomial or a sum of monomials (terms) Example: x3 + xy + y2

Classifying Polynomials Types of Polynomials: Monomial – a #, variable or the product of the two. Binomial – the sum/difference of two monomials. example: x2 + 2xy Trinomial – the sum/difference of three monomials. example: 5a – ab + c4 Polynomials with more than three terms do not have a special name.

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. 6x2 + 5 – 8x – 2x3 7x2 + 2x4 – 11

Add & Subtract Polynomials Adding Polynomials Hint: combine like terms! Make sure they have the same variable & exponent. (4xy + 7x2 – 6y) + (3xy + 2y – 5x2) We are NOT multiplying or dividing so… DON’T TOUCH THE EXPONENTS!!!!

Example 1: Find the sum of (4x2 + 3x - 7) + (x2 + 10) 2 ways to simplify: Horizontal Vertical (4x2 + 3x – 7) + (x2 + 10) (4x2 + 3x – 7) + ( x2 + 10)

Watch out for mixed up polynomials Watch out for mixed up polynomials! Example 2: Find the sum of (7 + 3x2 + 5xy) + (xy – 2x2 + 2)

Subtracting Polynomials Hint: distribute the subtraction sign then combine like terms! Example 3: Find the difference. (3x2 + 2x – 6) – (2x + x2 + 3)

Example 4: Find the difference. (5ab2 + 3ab) – (2ab2 + 4 – 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: Use the DISTRIBUTIVE PROPERTY! 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 Guess what: 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 The box method is more visual and can help you make sure that you have not missed multiplying any terms.

Box Method Example 7: (2x + 3)(5x + 8) Draw a box and write one binomial on the top and the other on the bottom. Multiply each pair of terms. If the terms are in standard form, we can use the peanut method!

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

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 – 3)2

Can we classify the polynomials below? Example 11) (x - 2)(x2 – 4x + 3)

We Meet and Greet, make sure everyone meets every term! Or… We Meet and Greet, make sure everyone meets every term!

Try this one by yourself… (3x + 7)(2x2 – x + 5)

(r – 2)(3r2 + 4r – 1) (4ab – 2a + 3)(a + b)(2a – 7) Examples 12 & 13: (r – 2)(3r2 + 4r – 1) (4ab – 2a + 3)(a + b)(2a – 7)

Example 14: Find the area of the rectangle below:

Challenge Question! Simplify: x2(x + 1) + 5x(x – 3) – 4(x + 10) Simplify: -4b(2b + 1) – 8(b2 + 2b – 2) Simplify: x2(x + 1) + 5x(x – 3) – 4(x + 10)

Homework Worksheet Work needs to be shown!!!