Polynomials Unit 5

Modeling Polynomials Polynomial:  An algebraic expression that contains one term or a sum of terms. The term(s) may contain variables (which will have whole number exponents).  And a term may be a number.

Modeling Polynomials 3x + 1 Is a polynomial. It contains a variable (whose exponent is 1) and numbers Since it is an expression, there is no equal sign

Modeling Polynomials 3x + 1 It has 2 terms A term is a number, or a variable, or the product of numbers and variables. Terms are separated by a + or a – Therefore 3x is one term and 1 is another term

Modeling Polynomials 3x + 1 In the term 3x the 3 is the numerical coefficient. This is the number in front of the variable It is the numerical factor of a term X is called the variable 1 is called the constant term There is no variable attached to this number

Modeling Polynomials We can classify a polynomial by the number of terms it has. Polynomials with 1, 2 or 3 terms have special names.

Modeling Polynomials Monomial has 1 term Binomial has 2 terms 5x or 9 or -2p2 Binomial has 2 terms 2c-5 or 2m2 + 3m or x + y Trinomial has 3 terms 2h2-6h+4 or x+y+z

Modeling Polynomials Just as numbers are the building block of arithmetic Polynomials are the building blocks of algebra! We use algebra tiles the same as base ten blocks but now we are representing polynomials

Try Some Use the algebra tiles provided Model each of the following:

5.1 – Modeling Polynomials Write an equation for each model. Is it a monomial, binomial or trinomial? A B

5.1 – Modeling Polynomials Use algebra tiles to model this polynomial: 4 – 2x2 + x What is the variable? How many terms are there? What are the coefficients? Are there any constants?

5.1 – Modeling Polynomials When is a number or expression NOT a polynomial? When a variable is in the denominator Square root of a variable X-2 X1/2 3/x A polynomial is one term (or the sum of terms) whose variables have whole-numbered exponents

5.1 – Modeling Polynomials Each part of a polynomial expression is called a term Terms begin with a + or a – (unless they are the first term) 2x2 + 4x + 3x + 2 – 8 ____ terms Simplify - ____ terms

5.1 – Modeling Polynomials We use monomial, binomial and trinomial to describe how many terms a simplified expression has Simplify where you can and classify each polynomial

5.1 – Modeling Polynomials For each term, the number that is multiplied by the variable is called the coefficient If there is no variable, the terms is called a constant A What are the coefficients and constants for each expression? B C

5.1 – Modeling Polynomials The degree of a term is determined by the exponent of the variable If there is no variable it has a degree of 0 The term 2x2 has a degree of 2 The term 2x has a degree of 1 The term 2 has a degree of 0

5.1 – Modeling Polynomials In your notebook write down the polynomials and identify their degree: x2 4x 3x6 8 4x3 - x12

5.1 – Modeling Polynomials Just like polynomials have names based on their number of terms they are also named based on their degree. Degree Name Constant 1 Linear 2 Quadratic 3 Cubic 4 Quartic 5 Quintic N >6 nth degree

5.1 – Modeling Polynomials In your notebook write down the polynomials and identify their names: x2 4x 3x6 8 4x3 - x12

5.1 – Modeling Polynomials The term with the greatest exponent determines the degree of the polynomial 2x3 + 4x2 – 3x + 8 is a _____ degree polynomial

5.1 – Modeling Polynomials Identify the degree of the following polynomials x2 + x – 3 -2x2 – 3 2x3 + 3x -2x4 -3x + 1 -3x +3 2x2 + x5 + 2x - 2

5.1 – Modeling Polynomials Polynomials should always be written from highest degree to lowest degree term Rearrange the following polynomials so that they are in order: X + 5 – x2 -2x2 – 3 + 4x 8 – 2x4 – 3x + x2

5.1 – Modeling Polynomials Math Practice – remember THIS is where your quiz and test questions come from Page 214 Questions – 4ace, 5, 6, 7, 8, 9ace, 11ace, 12, 13 ace, 14 and 18

Homework Review Put your name on the sticky note Write a polynomial that fits ALL the following criteria: 3 terms Complete polynomial has a degree of 2 One term has a degree of 1 Includes a constant term of 12 One negative term Two different coefficients

5.2 Like & Unlike Terms Any two opposite colored tiles of the same size has a sum of zero – these tiles are like terms Like Terms – terms that have the same variable, raised to the same exponent

5.2 Like & Unlike Terms

5.2 Like & Unlike Terms For the sake of visual clarity green tiles will be positive and red will be negative

5.2 Like & Unlike Terms

5.2 Like & Unlike Terms -x2 +3x - 4

5.2 Like & Unlike Terms

5.2 Like & Unlike Terms

5.2 Like & Unlike Terms 8x

5.2 Like & Unlike Terms What would be the perimeter of these shapes? A Will have a polynomial A – 6x +4 B – 6x+2

5.2 Like & Unlike Terms Simplifying a polynomial with two variables: Group like terms Combine (add/subtract) like terms Simplify the following: A - 11 – 5x2 + 3y + 4x – 5y +8y2 – 2x – 6 – 5y2 – 2x2 B - 11x – 2y2 – 3x2 + 4 – y + 10 – 4x – 3y – 5y – x

5.2 Like & Unlike Terms Math Practice – remember THIS is where your quiz and test questions come from Page 222 Questions 4, 5, 6, 7, 8ace, 11ace, 12ace, 13ace, 14ace, 17, 18, 19ac, 22

5.3 Adding Polynomials To add polynomials you collect like terms Three methods: Combine algebra tiles, group like tiles and cancel them out Add horizontally Add vertically

5.3 Adding Polynomials

5.3 Adding Polynomials

5.3 Adding Polynomials

5.3 Adding Polynomials

5.3 Adding Polynomials

5.3 Adding Polynomials Write a polynomial for the perimeter of this rectangle.

5.3 Adding Polynomials

5.3 Adding Polynomials Math Practice Page 228 Questions – 3, 5, 6, 8 (pick 2), 9 (pick 2), 10 (this question will for sure appear on a future quiz), 12

5.4 Subtracting Polynomials Just like adding polynomials, you collect the like terms There is only 1 MAJOR difference: when you drop the brackets you change the sign of each term after the subtraction sign

5.4 Subtracting Polynomials You must get the opposite of EVERY term in a polynomial.

5.4 Subtracting Polynomials When subtracting polynomials you must remember to ADD the OPPOSITE of every term.

5.4 Subtracting Polynomials How would you demonstrate this with algebra tiles? Model the following using tiles (-2x2 + x – 11) – (x2 – 3x + 2)

5.4 Subtracting Polynomials =y2 –8y+7

5.4 Subtracting Polynomials Math Practice Page – 234 – 236 Questions – 4, 7 (pick three), 8 (pick three), 9, 10 and 13

5.5 Multiplying and Dividing a Polynomial by a Constant We will only be multiplying and dividing polynomials by MONOMIALS Constants Contain 1 variable You will be expected to know symbolically, using area models and algebraically.

5.5 Multiplying and Dividing a Polynomial by a Constant

5.5 Multiplying and Dividing a Polynomial by a Constant

5.5 Multiplying and Dividing a Polynomial by a Constant

5.5 Multiplying and Dividing a Polynomial by a Constant

5.5 Multiplying and Dividing a Polynomial by a Constant Dividing Symbolically

5.5 Multiplying and Dividing a Polynomial by a Constant

5.5 Multiplying and Dividing a Polynomial by a Constant

5.5 Multiplying and Dividing a Polynomial by a Constant

5.5 Multiplying and Dividing a Polynomial by a Constant Math Practice Page 246 – 248 Questions – 3d, 4d, 5a, 9b, 10b, 11ace, 12, 13ace, 22d, 23d