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Polynomials Notes (Answers) Pure Math 10 Notes Unit 1: Polynomials Unit 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. Numerical Coefficient Monomial: - one term expression. Example: 5x2 Exponent Variable Binomial: - two terms expression. 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!! 1 Example: 9x −1 +12x 2 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 Classification Degree 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 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 Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 1. Unit 1: Polynomials Pure Math 10 Notes 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. (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 9x 2 + 4x d. Subtract This is the same as (9x2 + 4x) − (5x2 −7x) 5x 2 − 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. 24x7 y 4 z 5 75a3b4 a. (3x3y2) (7x2y4) b. c. 6x3 yz 5 25a5b3 24 x7 y 4 z 5 75 a3 b4 = (3)(7) (x3)(x2) (y2)(y4) = = 3 5 5 3 6 x y z 25 a b 3b = 21x5y6 = 4x4y3z0 ( z0 = 1 ) = 3a−2b or a 2 = 4x4y3 Page 2. Copyrighted by Gabriel Tang, B.Ed., B.Sc. Pure Math 10 Notes Unit 1: Polynomials (AP) Example 4: Find the area of the following ring. General Formula for Area of a Circle A = πr2 Inner Circle Radius = 2x Outer Circle Radius = (2x + 4x) = 6x Inner Circle Area: A = π (2x)2 A = π (4x2) 4x 4x A = 4πx2 Outer Circle Area: A = π (6x)2 A = π (36x2) A = 36πx2 Shaded Area = 36πx2 − 4πx2 Shaded Area = 32πx2 3-1 Homework Assignment Regular: pg. 102-103 #1 to 51, 55, 56 AP: pg. 102-103 #1 to 51, 53-57 Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 3. Unit 1: Polynomials Pure Math 10 Notes 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. 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 Page 4. Copyrighted by Gabriel Tang, B.Ed., B.Sc. Pure Math 10 Notes Unit 1: Polynomials Example 3: Find the shaded area of each of the followings. a. 5x + 4 b. 7x − 2 x + 2 3x + 1 x + 1 2x − 1 Shaded Area = Big Rectangle − Small Square x + 5 7x − 2 = (5x + 4) (2x − 1) − (x + 1) (x + 1) = (10x2 − 5x + 8x − 4) − (x2 + x + x + 1) x + 2 x + 2 = (10x2 + 3x − 4) − (x2 + 2x + 1) = 10x2 + 3x − 4 − x2 − 2x − 1 Shaded Area = 9x2 + x − 5 x + 5 (3x + 1) − (x + 2) = 2x − 1 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 3-3 Homework Assignment Regular: pg. 107-109 #1 to 77 (odd), 87, 88 AP: pg. 107-109 #2 to 84 (even) , 85, 87, 88, 91 Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 5. Unit 1: Polynomials Pure Math 10 Notes 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. (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) 2x + 1 = (9x2 + 12x + 4) − (4x2 − 4x + 1) 3x − 1 = 9x2 + 12x + 4 − 4x2 + 4x − 1 3x − 1 = 5x2 + 16x + 3 Volume = Length × Width × Height V = (3x − 1)2 (2x + 1) 3-4 Homework Assignment V = (9x2 − 3x − 3x + 1) (2x + 1) Regular: pg.112-113 #1 to 47 (odd), 49, 51, 54 (a, c, e, g), 55 (a, c, e, g), 56 V = (9x2 − 6x + 1) (2x + 1) AP: pg.112-113 #2 to 48 (even), 49 to 53, V = 18x3 + 9x2 − 12x2 − 6x + 2x + 1 54 (b, d, f, h), 55 (b, d, f, h), 56, 57 Volume = 18x3 − 3x2 − 4x + 1 Page 6. Copyrighted by Gabriel Tang, B.Ed., B.Sc. Pure Math 10 Notes Unit 1: Polynomials 3-6: Common Factors Common Factors can consist of two parts: a. Numerical GCF: - Greatest Common Factor of all numerical coefficients and constant. b. Variable GCF: - the lowest exponent of a particular variable. After obtaining the GCF, use it to divide each term of the polynomial for the remaining factor. Example 1: Factor the followings a. 3x2 + 6x + 12 b. 4a2b − 8ab2 + 6ab GCF = 3 GCF = 2ab = 3 (x2 + 2x + 4) = 2ab (2a − 4b + 3) Factor by Grouping (Common Brackets as GCF) a (c + d) + b (c + d) = (c + d) (a + b) Common Brackets Take common bracket out as GCF Example 2: Factor. a. 3x (2x − 1) + 4 (2x − 1) b. 2ab + 3ac + 4b2 + 6bc = (2x − 1) (3x + 4) = (2ab + 3ac) + (4b2 + 6bc) = a (2b + 3c) + 2b (2b + 3c) = (2b + 3c) (a + 2b) c. 3x2 − 6y2 + 9x − 2xy2 = (3x2 − 6y2) + (9x − 2xy2) = 3 (x2 − 2y2) + x (9 − 2y2) Brackets are NOT the same! We might have to first rearrange terms. Try again after rearranging terms! 3x2 + 9x − 2xy2 − 6y2 = (3x2 + 9x) − (2xy2 + 6y2) Switch Sign in Second Bracket! 2 We have put a minus sign in front of a new = 3x (x + 3) − 2y (x + 3) bracket! = (x + 3) (3x − 2y2) Copyrighted by Gabriel Tang, B.Ed., B.Sc. Page 7. Unit 1: Polynomials Pure Math 10 Notes Example 3: Find the area of the shaded region in factored form and as a polynomial. Shaded Area = Area of Square − Area of Circle 2 x Shaded Area = x2 − π x 2 πx 2 Shaded Area = x2 − (Polynomial Form) 4 π Shaded Area = x 2 1− (Factored Form) 4 Area of a Circle A = πr2 x Radius of Circle = 2 3-6 Homework Assignments Regular: pg. 120 #1 to 35 (odd), 38 to 43 AP: pg.120 #2 to 36 (even), 37 to 44 Page 8. Copyrighted by Gabriel Tang, B.Ed., B.Sc. Pure Math 10 Notes Unit 1: Polynomials 3-8: Factoring x2 + bx + c (Leading Coefficient is 1) x2 + bx + c What two numbers multiply to give c, but add up to be b? Example 1: Completely factor the followings. 2 Product of 6 2 a. x + 5x + 6 b. x − 3x − 10 Product of −10 1 6 −1 −6 −1 10 1 −10 2 3 = (x + 2) (x + 3) −2 − 3 = (x + 2) (x −5) −2 5 2 −5 sum of 5 sum of −3 Product of 15 c. a2 − 8a + 15 d. x2 − 7xy + 12y2 Product of 12 1 15 −1 −15 1 12 −1 −12 3 5 = (a − 3) (a − 5) −3 − 5 = (x − 3y) (x −4y) 2 6 −2 −6 3 4 −3 −4 sum of −8 sum of −7 e. x2y2 − 6xy − 16 Product of −16 f. 14 – 5w − w2 −1 16 1 −16 Rearrange in = (xy + 2) (xy − 8) −2 8 2 −8 = −w2 − 5w + 14 Descending Degree.
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