Lesson 5.1: Modelling Polynomials (P.210) (updated as of Oct. 2010) Focus: Model, write, and classify polynomials Complete Investigate (p.210) Algebra Tiles (only positives) Algebra Tiles (ans incl) Algebra Tiles (describe polynomial) Define the following: • Polynomial: ___________________________________________________________________________ i.e. non-example: • Terms: ____________________________________________________ i.e. • Coefficients: ________________________________________________ i.e. • Degree of a polynomial: _______________________________________ i.e. • Constant term: _______________________________________________ i.e. • Monomial: _____________________________________ i.e. • Binomial: ______________________________________ i.e. • Trinomial: ______________________________________ i.e. Example 1) Which of these polynomials can be represented by the same algebra tiles? a) 2 x 2 5 x 3 b) 5 3 p 2 p 2 c) 5m 3 2m 2 Example 2) Use algebra tiles to model each polynomial. Is the polynomial a monomial, binomial, or trinomial? Explain. a) 5x 2 b) 3b 2 5b 6 c) 4 2x Example 3) Which polynomial does each group of algebra tiles represent? Model A Model B Model C Assignment P.214 Lesson 5.2: Like Terms and Unlike Terms (p.217) Focus: Simplify polynomials by combining like terms These are all zero pairs: and and We can use zero pairs to simplify algebraic expressions. Example 1) Combining Like Tiles and Removing Zero Pairs Simplify this tile model. Write the polynomial that the remaining tiles represent. a) Polynomial ___________________________ b) Polynomial ___________________________ c) Polynomial ___________________________ Terms that can be represented by matching tiles are called __________ terms. • Examples of like terms: _______________________ __________________________ • Examples of Unlike terms: _____________________ ___________________________ We can simplify a ______________________ by adding the ______________________ of the _________ terms Example 2) Simplifying a Polynomial Symbolically Simplify: a) 6a 5 2a 6 c) 19 y 2 9 5 x 5 y x 2 4 y 2 2 x 2 b) 4 x 2 2 7 x 5 x 6 x 2 1 x d) 7 xy y 2 5x 2 3 yx y x 4 y 2 5x y 5.3 Adding Polynomials (p.225) Focus: Use different strategies to add polynomials Investigate (p.225) Example 1) Adding Polynomials with Algebra Tiles o Use algebra tiles to model 3s 2 2s 6 s 2 4s 2 concretely. o Draw the model pictorially below. o Write an addition sentence using symbols (algebraically). Pictorially Symbolically Example 2) Adding Polynomials Symbolically Add: 5 x 2 7 x 8 7 x 2 16 x 5 Example 3) Adding Polynomials Vertically Add: 4a 2 3a 7 7 a 2 5a Example 4) Write a polynomial for the perimeter of this rectangle. Simplify the polynomial. 3x 7 2x 1 Substitute to check your answer. Example 5) Adding Polynomials in Two Variables 3a 2 4a 5b 8ab 5b 2 6a 2 6ab 8b 9a 2 5.4 Subtracting Polynomials (p.231) Focus: Use different strategies to subtract polynomials. Example 1) Represent the expression 5 x 2 2 x 2 in 3 different ways, then simplify. Method 1: Using tiles (__________________) Method 2: Using tiles (__________________) Symbolically (algebraic) Example 2) Represent the expression 4 x 3x in 3 different ways, then simplify. Method 1: Using tiles (__________________) Method 2: Using tiles (__________________) Symbolically (algebraic) Example 3) Represent the expression 6 x 2 2 x 2 in 3 different ways, then simplify. Method 1: Using tiles (__________________) Symbolically (algebraic) Method 2: Using tiles (__________________) Example 4) Represent the expression 6 x 2 3x 4 in 3 different ways, then simplify. Method 1: Using tiles (__________________) Method 2: Using tiles (__________________) Symbolically (algebraic) Example 5) Represent the expression 3 x 2 2 x 7 2 x 2 3 x 2 in 3 different ways, then simplify. Method 1: Using tiles (__________________) Method 2: Using tiles (__________________) Symbolically (algebraic) Example 6) Simplify each algebraically. a) 5x 3x _____ d) 3x 2 6 x 4 x 2 3x 2 b) 2 x 2 3x 4 x 2 6 x ______________ __________ c) 3x 2 4 xy y 2 5 x 2 6 xy 7 y 2 e) 3a 2 2a 1 a 2 4a 3 ______________ ______________________ 5.5 Multiplying & Dividing a Polynomial by a Constant FOCUS: Use different strategies to multiply & divide a polynomial by a constant Discuss what 3 25 means? How can you represent this using a picture? Tiles Applet INVESTIGATE: (P.241) A. 2 3x D. 2 3x B. 3 2 x 1 E. 3 2 x 1 C. 2 2 x 2 x 4 F. 2 2 x 2 x 4 G. 9x 3 J. 9 x 3 H. 8x 12 4 K. I. 5x 8x 12 4 L. 5x 2 10 x 20 5 2 10 x 20 5 Example 1) Multiplying a Binomial and a Trinomial by a Constant Determine each product pictorially and symbolically: a) 2 3m 4 Pictorially Method 1 Method 2 Symbolically Method 3 b) 3 n 2 2n 1 Pictorially Symbolically NOTE: Multiplication & division are ________________ operations. To divide a polynomial by a constant, we ______________ the process of multiplication. Representing division using a Model 6x 3 3 Algebra Tiles Area Model Algebraically Example 2) Dividing a Binomial and a Trinomial by a Constant a) b) 4 x 2 12 4 Method 1: Tiles Method 2: Algebraic (breaking into terms) 3m2 15mn 21n 2 3 Method 1: Division = backwards multiplication Method 2: Algebraic (breaking into terms) 5.6 Multiplying & Dividing a Polynomial by a Monomial (p.249) FOCUS: use different strategies to multiply and divide a polynomial by a monomial Example 1) Multiplying a Binomial by a Monomial Represent each expression using tiles, then complete the product. A) 2 x 4 x Method 1: Tiles Method 2: Area model Method 3: Algebraic B) 3x 2 x 4 Method 1: Tiles Method 2: Area model Method 3: Algebraic (using the distributive property) C) 4c 2c 3 Method 1: Tiles Method 2: Distributive property Example 2) A) 2a 5a B) 4b 3b 2 C) 3c 5c 1 NOTE: As before, multiplication & division are ________________ operations. To divide a polynomial by a monomial, we ______________ the process of multiplication. 6x2 Example 3) Draw . Simplify. 2x 6 w2 9 w Example 4) Draw . Simplify. 3w Example 5) Dividing a Monomial and a Binomial by a Monomial Determine the quotient of each. 12m 2 36k 18k 2 A) D) 3m 6k B) 3g 2 9 g 3g E) 18 f 2 12 f C) 6f 24d 2 8d 4d