Polynomials CHAPTER 5 Chapter MODELING POLYNOMIALS POLYNOMIALS 3y 4 4y + 6 2x 3 1 2y + 3x 7 3 terms, degree of 4 2 terms, degree of 3 3 terms, degree of 1 TERMS Constant A number Degree of 0 Linear x Degree of 1 Quadratic x 2 Degree of 2 Cubic x 3 Degree of 3 MIX AND MATCH GAME POLYNOMIAL TILES So, what would x 2 + x 3 look like? TRY IT 2x 2 3x + 1 5m m 2 EXAMPLE What is the polynomial representation of this model? 2 big yellow squares +2x 2 8 red lines 8x 2 small yellow squares +2 2x 2 8x + 2 COMBINING LIKE TERMS They cancel! So you can get rid of one yellow and one red big square. Leftover, we have: 1 big yellow square x 2 2 red lines 2x 4 little yellow squares +4 x 2 2x + 4 PG , # 5, 8, 9, 12, 13, 20 Chapter LIKE TERMS AND UNLIKE TERMS 5.3 ADDING POLYNOMIALS EXAMPLE = = x 2 3x x 2 2x x 2 2x = x 2 4x - 1 EXAMPLE Use algebra tiles to simplify the polynomial 4n 2 1 3n 3 + 5n 2n 2. 4n 2 1 3n 3 + 5n 2n 2 = 2n 2 + 2n 4 This process is called combining like terms. It helps us find the simplified form of a polynomial. SIMPLIFIED FORM We can tell a polynomial is in simplified form when: its algebra tile model uses the fewest tiles possible its symbolic form contains only one term of each degree and no terms with a zero coefficient Terms that can be represented by algebra tiles with the same size and shape are like terms. TRY IT Simplify, either by using algebra tiles or by finding like terms: 14x 2 x x 25x 2 = (14 25)x 2 + ( )x + (3 11) = 11x x 8 EXAMPLE Simplify: 4xy y 2 3x 2 + 2xy x 3y 2 When you have two variables in an expression, the like terms need to be the same for both variables. 4xy y 2 3x 2 + 2xy x 3y 2 = 3x 2 x + 6xy 4y 2 EXAMPLE Find the sum of 3x 2 + 2x + 4 and 5x 2 + 3x 5 (3x 2 + 2x + 4) + (5x 2 + 3x 5) = (3 5)x 2 + (2 + 3)x + (4 5) = 2x 2 + 5x 1 TRY IT Simplify: (2a 2 + a 3b 7ab + 3b 2 ) + (4b 2 + 3ab + 6b 5a + 5a 2 ) = (2 + 5)a 2 + (1 5)a + (3 + 6)b + (7 + 3)ab + (3 4)b 2 = 7a 2 4a + 3b 4ab b 2 POLYNOMIAL TILES 1.Split into pairs. 2.Put your polynomial tiles in a bag. 3.Construct a table to record your work, and make sure to include both the algebra tile model and the symbolic record (the algebraic expression) 4.Take a handful out and sketch them. 5.Remove the zero pairs and sketch again. 6.Repeat 4 times. Algebra Tile ModelAlgebra Tiles without the Zero Pairs Symbolic Record Independent Practice PG # 6, 9, 11, 13, 14, 19, 21. PG # 9, 10, 12, 14, 17, 18. Chapter SUBTRACTING POLYNOMIALS SUBTRACTING What happens when you subtract a positive number? What happens when you subtract a negative number? EXAMPLE 3x 2 4x (2x 2 6x) = 3x 2 4x 2x 2 + 6x = x 2 + 2x TRY IT Subtract: (2a 2 + a 1) (a 2 3a + 2) (2a 2 + a 1) (a 2 3a + 2) = 2a 2 + a 1 a 2 + 3a 2 = 3a 2 + 4a 3 Independent Practice PG , # 4, 7, 8, 9, 12, 13, 15, 16, 17