Polynomials Unit 5 Modeling Polynomials Polynomial An algebraic

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 • 3 x + 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 • 3 x + 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 3 x is one term and 1 is another term

Modeling Polynomials • 3 x + 1 – In the term 3 x 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 5 x or 9 or -2 p 2 • Binomial has 2 terms 2 c-5 or 2 m 2 + 3 m or x + y • Trinomial has 3 terms 2 h 2 -6 h+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– – – What is the variable? How many terms are there? What are the coefficients? Are there any constants? 2 2 x +x

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 X 1/2 3/x

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) terms 2 x 2 + 4 x + 3 x + 2 – 8 Simplify ____ terms ____

5. 1 – Modeling Polynomials • We use monomial, binomial and trinomial to describe how many terms a simplified expression has

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 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 2 x 2 has a degree of 2 The term 2 x 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: x 2 4 x 3 x 6 8 4 x 3 - x 12

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 0 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: x 2 4 x 3 x 6 8 4 x 3 - x 12

5. 1 – Modeling Polynomials • The term with the greatest exponent determines the degree of the polynomial 2 x 3 + 4 x 2 – 3 x + 8 is a _____ degree polynomial

5. 1 – Modeling Polynomials • Identify the degree of the following polynomials x 2 + x – 3 -2 x 2 – 3 -2 x 4 -3 x + 1 -3 x +3 2 x 3 + 3 x 2 x 2 + x 5 + 2 x - 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 – x 2 -2 x 2 – 3 + 4 x 8 – 2 x 4 – 3 x + x 2

5. 1 – Modeling Polynomials • Math Practice – remember THIS is where your quiz and test questions come from • Page 214 • Questions – 4 ace, 5, 6, 7, 8, 9 ace, 11 ace, 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 • What would be the perimeter of these shapes? A B

5. 2 Like & Unlike Terms • Simplifying a polynomial with two variables: 1. Group like terms 2. Combine (add/subtract) like terms Simplify the following: A - 11 – 5 x 2 + 3 y + 4 x – 5 y +8 y 2 – 2 x – 6 – 5 y 2 – 2 x 2 B - 11 x – 2 y 2 – 3 x 2 + 4 – y + 10 – 4 x – 3 y – 5 y – 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, 8 ace, 11 ace, 12 ace, 13 ace, 14 ace, 17, 18, 19 ac, 22

5. 3 Adding Polynomials • To add polynomials you collect like terms • Three methods: 1. Combine algebra tiles, group like tiles and cancel them out 2. Add horizontally 3. Add vertically

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 (-2 x 2 + x – 11) – (x 2 – 3 x + 2)

5. 4 Subtracting Polynomials

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 Dividing Symbolically

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 – 3 d, 4 d, 5 a, 9 b, 10 b, 11 ace, 12, 13 ace, 22 d, 23 d