Transcription of Unit 1: Polynomials
1 Pure Math 10 Notes unit 1: Polynomials Copyrighted by Gabriel Tang, , Page 1. ExponentUnit 1: Polynomials 3-1: Reviewing Polynomials Expressions: - mathematical sentences with no equal sign. Example: 3x + 2 Equations: - mathematical sentences that are equated with an equal sign. Example: 3x + 2 = 5x + 8 Terms: - are separated by an addition or subtraction sign. - each term begins with the sign preceding the variable or coefficient. Monomial: - one term expression. Example: Binomial: - two terms expression.
2 Example: 5x2 + 5x Trinomial: - three terms expression. Example: x2 + 5x + 6 Polynomial: - many terms (more than one) expression. All Polynomials must have whole numbers as exponents!! Example: 211129xx+ is NOT a polynomial. Degree: - the term of a polynomial that contains the largest sum of exponents Example: 9x2y3 + 4x5y2 + 3x4 Degree 7 (5 + 2 = 7) Example 1: Fill in the table below. Polynomial Number of Terms ClassificationDegree Classified by Degree 9 1 monomial 0 constant 4x 1 monomial 1 linear 9x + 2 2 binomial 1 linear x2 4x + 2 3 trinomial 2 quadratic 2x3 4x2 + x + 9 4 polynomial 3 cubic 4x4 9x + 2 3 trinomial 4
3 Quartic Like Terms: - terms that have the same variables and exponents. Examples: 2x2y and 5x2y are like terms 2x2y and 5xy2 are NOT like terms 5x2 Numerical CoefficientVariableUnit 1: Polynomials Pure Math 10 Notes Page 2. Copyrighted by Gabriel Tang, , To Add and Subtract Polynomials : Combine like terms by adding or subtracting their numerical coefficients. Example 2: Simplify the followings. a. 3x2 + 5x x2 + 4x 6 b. (9x2y3 + 4x3y2) + (3x3y2 10x2y3) = 3x2 + 5x x2 + 4x 6 = 9x2y3 + 4x3y2 + 3x3y2 10x2y3 = 2x2 + 9x 6 = x2y3 + 7x3y2 c.
4 (9x2y3 + 4x3y2) (3x3y2 10x2y3) = 9x2y3 + 4x3y2 3x3y2 + 10x2y3 (drop brackets and switch signs in the bracket that had sign in front of it) = 19x2y3 + x3y2 d. Subtract xxxx754922 + This is the same as (9x2 + 4x) (5x2 7x) = 9x2 + 4x 5x2 + 7x = 4x2 + 11x To Multiply and Divide Monomials: Multiply or Divide (Reduce) Numerical Coefficients. Add or Subtract exponents of the same variable according to basic exponential laws. Example 3: Simplify the followings. a. (3x3y2) (7x2y4) b. 53547624yzxzyx c. 35432575baba = (3)(7) (x3)(x2) (y2)(y4) = 55437624zzyyxx = 34532575bbaa = 21x5y6 = 4x4y3z0 ( z0 = 1 ) = 3a 2b or 23ab = 4x4y3 Pure Math 10 Notes unit 1: Polynomials Copyrighted by Gabriel Tang, , Page 3.
5 (AP) Example 4: Find the area of the following ring. General Formula for Area of a Circle A = r2 Inner Circle Radius = 2x Outer Circle Radius = (2x + 4x) = 6x Inner Circle Area: A = (2x)2 A = (4x2) A = 4 x2 Outer Circle Area: A = (6x)2 A = (36x2) A = 36 x2 Shaded Area = 36 x2 4 x2 Shaded Area = 32 x2 3-1 Homework Assignment Regular: pg. 102-103 #1 to 51, 55, 56 AP: pg. 102-103 #1 to 51, 53-57 4x 4x unit 1: Polynomials Pure Math 10 Notes Page 4.
6 Copyrighted by Gabriel Tang, , 3-3: Multiplying Polynomials To Multiply Monomials with Polynomials Example 1: Simplify the followings. a. 3 (2x2 4x + 7) b. 2x (3x2 + 2x 4) = 3 (2x2 4x + 7) = 2x (3x2 + 2x 4) = 6x2 12x + 21 = 6x3 + 4x2 8x c. 3x (5x + 4) 4 (x2 3x) d. 8 (a2 2a + 3) 4 (3a2 + 7) (only multiply = 3x (5x + 4) 4 (x2 3x) the brackets = 8 (a2 2a + 3) 4 (3a2 + 7) right after the = 15x2 + 12x 4x2 + 12x monomial) = 8a2 16a + 24 4 3a2 7 = 11x2 + 24x = 5a2 16a + 13 To Multiply Polynomials with Polynomials Example 2: Simplify the followings.
7 A. (3x + 2) (4x 3) b. (x + 3) (2x2 5x + 3) = (3x + 2) (4x 3) = (x + 3) (2x2 5x + 3) = 12x2 9x + 8x 6 = 2x3 5x2 + 3x + 6x2 15x + 9 = 12x2 x 6 = 2x3 + x2 12x + 9 c. 3 (x + 2) (2x + 3) (2x 1) (x + 3) d. (x2 2x + 1) (3x2 + x 4) = 3 (x + 2) (2x + 3) (2x 1) (x + 3) = (x2 2x + 1) (3x2 + x 4) = 3 (2x2 3x + 4x 6) (2x2 + 6x x 3) = 3x4 + x3 4x2 6x3 2x2 + 8x + 3x2 + x 4 = 3 (2x2 + x 6) (2x2 + 5x 3) = 6x2 + 3x 18 2x2 5x + 3 = 3x4 5x3 3x2 + 9x 4 = 4x2 2x 15 Pure Math 10 Notes unit 1: Polynomials Copyrighted by Gabriel Tang, , Page 5.
8 Example 3: Find the shaded area of each of the followings. a. b. Shaded Area = Big Rectangle Small Square = (5x + 4) (2x 1) (x + 1) (x + 1) = (10x2 5x + 8x 4) (x2 + x + x + 1) = (10x2 + 3x 4) (x2 + 2x + 1) = 10x2 + 3x 4 x2 2x 1 Shaded Area = 9x2 + x 5 Total Area = Top Rectangle + Bottom Rectangle = (7x 2) (x + 2) + (2x 1) (x + 5) = (7x2 + 14x 2x 4) + (2x2 + 10x x 5) = (7x2 + 12x 4) + (2x2 + 9x 5) = 7x2 + 12x 4 + 2x2 + 9x 5 Total Area = 9x2 + 21x 9 x + 1 5x + 4 2x 17x 2 x + 2 3x + 1 x + 5 7x 2 x + 2 (3x + 1) (x + 2)=2x 1x + 5 x + 2 3-3 Homework Assignment Regular: pg.
9 107-109 #1 to 77 (odd), 87, 88 AP: pg. 107-109 #2 to 84 (even) , 85, 87, 88, 91 unit 1: Polynomials Pure Math 10 Notes Page 6. Copyrighted by Gabriel Tang, , 3-4: Special Products (x + y)2 = (x + y) (x + y) (x + y)2 is NOT x2 + y2 (x + y)3 = (x + y) (x + y) (x + y) Example 1: Simplify the followings. a. (2x + 3)2 b. (x 2)3 = (2x + 3) (2x + 3) = (x 2) (x 2) (x 2) = 4x2 + 6x + 6x + 9 = (x 2) (x2 2x 2x + 4) = 4x2 + 12x + 9 = (x 2) (x2 4x + 4) = x3 4x2 + 4x 2x 2 + 8x 8 = x3 6x2 + 12x 8 c.
10 (3x + 2)2 (2x 1)2 Example 2: Find the volume of the box below. = (3x + 2) (3x + 2) (2x 1) (2x 1) = (9x2 + 6x + 6x + 4) (4x2 2x 2x + 1) = (9x2 + 12x + 4) (4x2 4x + 1) = 9x2 + 12x + 4 4x2 + 4x 1 = 5x2 + 16x + 3 Volume = Length Width Height V = (3x 1)2 (2x + 1) V = (9x2 3x 3x + 1) (2x + 1) V = (9x2 6x + 1) (2x + 1) V = 18x3 + 9x2 12x2 6x + 2x + 1 Volume = 18x3 3x2 4x + 1 2x + 1 3x 1 3x 1 3-4 Homework Assignment Regular: #1 to 47 (odd), 49, 51, 54 (a, c, e, g), 55 (a, c, e, g), 56 AP.
