Warm-Up 1. f( g(x)) = ____ for g(x) = 2x + 1 and f(x) = 4x , if x = 3 2. (f + g)(x) = ____ for g(x) = 3x2+ 2x and f(x) = 3x + 1 3. (f/g)(x) = ______ for f(x) = 3x2 +3x and g(x) = 3x 4. (f/g)(x) = ______ for f(x) = 3x2 +3x and g(x) = 3x, for x = 3

Objective Students will be able to simplify like terms when adding, subtracting, multiplying and dividing polynomials and solving equations. Students will be able to

Vocabulary and background Monomial – An expression like 5x is called a monomial. A monomial is an integer, a variable, or a product of integers or variables. Coefficient – The numerical part of a monomial Like Terms – When monomials are the same or differ only by their coefficients they are called like terms Exponent – tells how many times a number, called the base, is used as a factor. Powers – numbers that are expressed using exponents are called powers. Multiplicative Inverse/Reciprocals – Two numbers whose product is 1. For example, 2 * ½ = 1, and 2/3 * 3/2 = 1. Additive Inverse – An integer and its opposite are called additive inverses of each other. The sum of an integer and its additive inverse is zero. For example x + (-x) = 0 and 3 + (-3) = 0. Polynomial – An algebraic expression that contains one or more monomials is called a polynomial; two term polynomials are called bionomical, and three term polynomials are called trinomial.

Adding Polynomials Rule – Zero pair is formed by pairing one tile with its opposite or by adding additive inverses. Simplify (2x2 + 2x – 4) + (x2 + 3x + 6) 1. Distribute the 1 + 2x2 + 2x – 4 + x2 + 3x + 6 2. Group like terms. + + 2x2+ x2 + 3x + 2x -4 + 6 3. Add or subtract coefficients 3x2 + 5x+2 4. Answer 1 x2 x2 x2 1 1 x x x 1 x x -1 -1 1 1 -1 -1 x2 x2 x2 x x x x x 1 -1 1 -1 1 1 -1 1 -1 1

Subtracting Polynomials x2 -x2 x -x 1 -1 Rule – Zero pair is formed by pairing one tile with its opposite or by adding additive inverses. Simplify (2x2 + 2x – 4) – (x2 + 3x + 6) 1. Distribute the -1 - 2x2 + 2x – 4 - x2 - 3x - 6 2. Group like terms,. + + 2x2 - x2 + 2x - 3x -4 - 6 3. Add or subtract coefficients x2 + (-x) + (-10) or x2- x – 10 4. Answer 1 1 x2 x2 x2 1 1 x x x 1 x x -1 -1 1 1 -1 -1 x2 x2 -x2 x x -x -x -x -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

Increasing the Challenge Add. ( 6y – 5r) + (2y + 7r) (6x2 + 15x – 9) + (5 – 8x – 8x2 ) Subtract 1. (4x2 + 7x + 4) – (x2 + 2x + 1) 2. (5x2y2 + 11xy – 9 ) – ( 9x2y2 – 13xy + 6)

Increasing the Challenge Add. 1. ( 6y – 5r) + (2y + 7r) 6y – 5r + 2y + 7r Distributive Property / Distribute 1 to remove the parentheses 6y + 2y -5r + 7r Group like terms and simplify 8y + 2r 2. (6x2 + 15x – 9) + (5 – 8x – 8x2 ) 6x2 + 15x – 9 + 5 – 8x – 8x2 Distributive Property / Distribute 1 to remove the parentheses 6x2 – 8x2 + 15x – 8x– 9 + 5 Group like terms and simplify – 2x2 – 7x– 4 Subtract 1. (4x2 + 7x + 4) – (x2 + 2x + 1) 4x2 + 7x + 4 – x2 - 2x – 1 Distributive Property / Distribute -1 to remove the parentheses 4x2 – x2 + 7x - 2x + 4 – 1 Group like terms and simplify 3x2 + 5x + 3 2. (5x2y2 + 11xy – 9 ) – ( 9x2y2 – 13xy + 6) 5x2y2 + 11xy – 9 – 9x2y2 + 13xy – 6 Distributive Property / Distribute -1 to remove the parentheses 5x2y2 – 9x2y2 + 11xy + 13xy – 9 – 6 Group like terms and simplify – 4x2y2 + 23xy – 15

Multiplying and Dividing Polynomials

Powers of Monomials Ex. a5 * a3 = a5-3 = a2 Product of Powers – You can multiply powers that have the same base by adding their exponents. For any number a and positive integers m and n. am * an= a m+n Ex. a5 * a3 = a5-3 = a2 Quotient of Powers – You can divide powers that have the same base by subtracting their exponents. For any nonzero number a and whole numbers m and n. Ex. a4/a2 = a4-2 = a2 Negative Exponents – For any number a and any integer n, a-n = Ex. 5-2 = 1/52 = 1/25

Try This 1. Which expression is equivalent to (4x2 + 8x + 4) – (x2 + 2x + 2). a. 3x2 + 6x + 2 b. 3x2 + 10x + 6 c. 3x2 - 6x + 2 d. 5x2 + 6x + 2 2 Which expression is equivalent to . a. 2x3y6z3 b. 2xy2z2 c. 4y2z2 d. 4yz3

Try This 1. Which expression is equivalent to (4x2 + 8x + 4) – (x2 + 2x + 2). a. 3x2 + 6x + 2 b. 3x2 + 10x + 6 c. 3x2 - 6x + 2 d. 5x2 + 6x + 2 2 2. Which expression is equivalent to . a. 2x3y6z3 b. 2xy2z2 c. 4y4z2 4yz3 Explanation: (16/4)( x6-6)(y8-4)(z4-2) = 4y4z2 Quotient of Powers 4x2 + 8x + 4 – x2 - 2x – 2 4x2 – x2 + 8x - 2x + 4 – 2 3x2 + 6x+ 2 2 = 16x6y8z4 = 4y4z2 4x6y4z2

Order of Operations

More Practice 1) −9 − 6(−v + 5) 2) −10(−8x + 9) − 8x 3) 1 + 4(2 − 3k) 4) −8v + 6(10 + 6v) 5) 7(1 + 9v) − 8(−5v − 6) 6) −10(x − 7) − 7(x + 2) 7) −2(−6x − 9) − 4(x + 9) 8) 9(7k + 8) + 3(k − 10)

(7x + 2)(5x+1) 5x 1 7x 2 = 35x2+7x+10x +2 = 35x2+17x +2 Multiplying Polynomials (7x + 2)(5x+1) 5x 1 7x 2 = 35x2+7x+10x +2 = 35x2+17x +2 7x *5x 7x *1 2 *5x 2 * 1

Guided Practice FOIL Method

Column Form Product of 4x3-32x2+0x +36 * 4 Product of 3x4-24x3+0x +27x * 3x Sum of the product

Special Products Square of a Difference (a-b)2 = (a-b)(a-b) = a2-2ab + b2 Find (r- 6)2. Difference of Squares (a + b)(a – b) = (a-b)(a + b) = (a2 – b2) Find (m -2n)(m + 2n).

Dividing. 1. = 4 𝑥 2 𝑦 6 𝑧 5 )(8𝑥 𝑦 10 𝑧 2 𝑥 2 𝑦 3 𝑧 12 = 4 𝑥 2 𝑦 6 𝑧 5 )(8𝑥 𝑦 10 𝑧 2 𝑥 2 𝑦 3 𝑧 12 = 8 𝑥 7 𝑦 6 𝑧 5 2 𝑥 2 𝑦 3 𝑧 9

Long Division

Individual Practice

Guided Practice

Summary

Homework