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APRIL 4, 2017 UNIT 2: FACTORS AND PRODUCTS SECTION 3.3: COMMON FACTORS OF A POLYNOMIAL M. MALTBY INGERSOLL NUMBERS, RELATIONS AND FUNCTIONS 10 1 WHAT'S THE POINT OF TODAY'S LESSON? We will begin working on the NRF 10 Specific Curriculum Outcome (SCO) "Algebra and Numbers 5" OR "AN5" which states: "Demonstrate an understanding of common factors and trinomial factoring." 2 What does THAT mean??? SCO AN5 means that we will: * determine the common factors in the terms of a polynomial and express the polynomial in factored form * factor a polynomial that is a "difference of squares" and explain why it is a special case of trinomial factoring where b = 0 * identify and explain errors in a polynomial factorization * factor a polynomial and verify by multiplying the factors * explain, using examples, the relationship between multiplication and factoring of polynomials * generalize and explain strategies used to factor a trinomial * express a polynomial as a product of its factors 3 WARMUP: EXPAND: (5x 7)3 SOLUTION: (5x 7)3 = (25x2 70x + 49)(5x 7) = 125x3 175x2 350x2 + 490x + 245x 343 = 125x3 525x2 + 735x 343 4 HOMEWORK QUESTIONS??? (pages 186 / 187, #6(i, iii, v, vii) TO #11 and #18) 5 IN ARITHMETIC: IN ALGEBRA: Multiply factors to form a product: Expand an expression to form a product: (4)(7) = 28 3(2 5a) = 6 15a Factor a number by writing it as a product of factors: Factor a polynomial by writing it as a product of factors: 28 = (4)(7) 6 15a = 3(2 5a) (EXPANDING and FACTORING are inverse processes.) 6 MULTIPLYING MONOMIALS BY BINOMIALS: Last week, we built gradually on what we knew in grade 9. We EXPANDED expressions to form products. For example: 3(x + 4) = 3(x) + 3(4) (We use the DISTRIBUTIVE PROPERTY here.) = 3x + 12 7 FACTORING BINOMIALS: For example, FACTOR the following binomial: 3x + 12 = (3)(x) + (3)(2)(2) = 3(x + 4) (This is a PRODUCT; it is the answer to a multiplication.) (Prime factorization was used here; GCF = 3.) (This is the FACTORED form of the product using the GCF of 3.) To check that your answer is correct, EXPAND! 3(x + 4) = 3x + 12 8 FACTORING BINOMIALS: This is how factoring actually works without prime factorization: 3x + 12 = 3(x + 4) (Ask yourself: What is the GCF of 3 and 12? 3!) (Mentally divide 3x by 3 and 12 by 3.) 9 MULTIPLYING MONOMIALS BY BINOMIALS: Again, we built gradually on what we knew in grade 9... For example: 6x(2x + 3) = (6)(2)(x)(x) + (6)(3)(x) = 12x2 + 18x 10 FACTORING BINOMIALS: Now, let's FACTOR the following binomial: 12x2 + 18x = (2)(2)(3)(x)(x) + (2)(3)(3)(x) = 6x(2x + 3) (GCF = 6x) Check: 6x(2x + 3) = 12x2 + 18x 11 FACTORING BINOMIALS: This is how factoring actually works without prime factorization: 12x2 + 18x = 6x(2x + 3) (Ask yourself: What is the GCF of 12 and 18? 6! What is the GCF of x2 and x? x! GCF = 6x) 2 (Mentally divide 12x by 6x and 18x by 6x.) 12 FACTORING BINOMIALS: We have to deal with negative terms when factoring binomials. For example, FACTOR: 3y2 9y = 3y(y 3) (Ask yourself: What is the GCF of 3 and 9? 3! What is the GCF of y2 and y? y! GCF = 3y) 2 (Mentally divide 3y by 3y and 9y by 3y.) Check: 3y(y 3) = 3y2 9y 13 FACTORING BINOMIALS: We have to deal with negative terms when factoring binomials. NOTE: We ensure that the first term inside the brackets is POSITIVE. For example, FACTOR: 8a2 12a = 4a(2a + 3) (Ask yourself: What is the GCF of 8 and 12? 4! What is the GCF of a2 and a? a! Now, to ensure a positive first term inside the brackets, use a GCF = 4a) 2 (Mentally divide 8a by 4a and 12a by 4a.) Check: 4a(2a + 3) = 8a2 12a 14 MULTIPLYING MONOMIALS BY TRINOMIALS: We also learned this process last week. For example: 6(x + 3y + 2) = (6)(x) + (6)(3)(y) + (6)(2) = 6x + 18y + 12 15 FACTORING TRINOMIALS: For example, FACTOR: 6x + 18y + 12 = 6(x + 3y + 2) (Ask yourself: What is the GCF of 6, 18 and 12? 6! (Mentally divide 6x by 6, 18y by 6 and 12 by 6.) Check: 6(x + 3y + 2) = 6x + 18y + 12 16 FACTORING TRINOMIALS: For example, FACTOR: 18a2 27a + 36ab (Ask yourself: What is the GCF of 18, 27 and 36? 9! What is the GCF of a2, a and ab? a! GCF = 9a) = 9a(2a 3 + 4b) 2 (Mentally divide 18a by 9a, 27a by 9a and 36ab by 9a.) Check: 9a(2a 3 + 4b) = 18a2 27a + 36ab 17 FACTORING TRINOMIALS: For example, FACTOR: 12x3y 20xy 2 16x2y2 = 4xy(3x 2 + 5y + 4xy) Check: 4xy(3x2 + 5y + 4xy) = 12x3y 20xy2 16x2y2 18 FACTORING TRINOMIALS: For example, FACTOR: 20c4d 30c3d2 25c2d = 5c2d(4c2 + 6cd + 5) Check: 5c2d(4c2 + 6cd + 5) = 20c4d 30c3d2 25c2d 19 CONCEPT REINFORCEMENT: FPCM 10: Page 155: Page 156: #8, #10 & #14 #16 20