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Flowchart to factor Two terms ? Three Terms? Four or More Terms? Factor out the Great Common Factor Can be the expression written as A 2 – B 2 or A+ B 3 or A 3 – B 3 ? YES NO YES NO YES Can the expression be written as ax 2 +bx+c? Can you factorize the expression by grouping ? YES Factor NO END YES perfect square? YES NO product of two binomials? YES NO END NO END Ex 2 Ex 3 Ex 4 Ex 5 Ex 1 END NO

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Ex 1: Greatest Common Factor (GCF) Ex 1a: Factor 6ab+8ac+4a Look for the GCF 6ab+8ac+4a = 2 ·3 ab + 2 · 4ac + 2 ·2a Take out the GCF = 2a ( ) Working on the parenthesis = 2a ( 3 b + 4 c + 2 ) Answer: 6ab+8ac+4a = 2a ( 3b + 4c + 2) Ex 1b: Factor 5x 2 y(2a–3) –15xy(2a-3) Look for the GCF 5x 2 y(2a–3)–15xy(2a-3) = 5xxy(2a-3)– 35xy(2a-3) Take out the GCF = 5xy(2a-3)( ) Working on the parenthesis = 5xy(2a-3)(x - 3 ) Answer: 5x 2 y(2a – 3) –15xy(2a - 3) = 5xy(2a - 3)( x - 3 ) Return to Flowchart

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Ex 2: Factoring a Binomial Ex 2a: Factor 3x 3 – 12xy 2 Taking the GCF out 3x 3 – 12xy 2 = 3x ( x 2 – 4y 2 ) Factoring as difference of squares Ex 2b: Factor 16x x 2 Taking the GCF out 16x x 2 = 2x 2 (8x ) Working on the parenthesis = 2x 2 ( [2x] 3 + [3] 3 ) Answer: 3x 3 – 12xy 2 = 3x (x+2y(x -2y) = 3x (x+2y)(x -2y) = 2x 2 (2x+ 3)[ (2x) 2 –(2x)3 +(3) 2 ] Simplifying = 2x 2 (2x+ 3 )(4x 2 - 6x + 9) Answer : 16x 5 – 54x 2 = 2x 2 (2x + 3)(4x 2 – 6x + 9) Factoring as difference of cubes Return to Flowchart

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Ex 3: Factoring as Perfect Square Ex 3a: Factor 63x 4 – 210x x 2 Taking GCF out 63x 4 – 210x x 2 = 7x 2 (9x 2 – 30x + 25) Check for perfect square ( )2)2 Take square root first and last terms = 7x 2 (3x – 5 )2)2 Answer: 63x 4 – 30x x 2 = 7x 2 (3x – 5 )2)2 Ex 3b: Factor 3x 3 y 2 +18x 2 y +27x Taking the GCF out 3x 3 y x 2 y +27x = 3x (x 2 y 2 + 6xy + 9) Check for perfect square ( )2)2 Take square root first and last terms = 3x ( xy + 3 )2)2 Middle term’s check 2(xy)3 = 6xy OK! Middle term’s check 2(3x)5 = 30x OK! Answer: 3x 3 y 2 +18x 2 y + 27 =3x (xy + 5) 2 Return to Flowchart

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Ex 4: Factor as Product of Two Binomials Simple Case Ex 4a : Factor 3x 3 – 21x x Taking GCF out 3x 3 – 21x x = 3x (x 2 – 7x+ 10) Trinomial is not a perfect square Let’s try to factor as product of two binomials = 3x(x + a )(x + b ) Where a·b = 10 & a + b = - 7. Since a·b is positive, a & b should have same sign, and since ab = - 7 both should be negative. Possibilities for a, b are 1,10 or 2,5. Let’s check x - 1 x - 10 x - 2 x - 5 So, a = -2 and b = - 5 3x 3 – 21x x = 3x (x -2)(x –5) Answer : 3x 3 – 21x x = 3x (x - 2)(x - 5) - x - 10x -11x -2x -5x -7x NO! OK! Return to Flowchart

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Ex 4: Simple Case Continued … Ex 4b: Factor 4x 5 – 20x x Taking the GCF out 4x 5 – 20x x = 4x (x 4 – 5x ) Trinomial is not a perfect square ! Factoring as product of two binomials = 4x (x 2 – 1) (x 2 – 4) Factoring as difference of squares =4x (x+1)(x–1)(x+2)(x–2) Answer: 4x 5 – 20x 3 +16x = 4x (x+1)(x–1)(x+2)(x–2) Ex 4c: Factor 128 x 7 – 2x Taking the GCF out … 128 x 7 – 2x = 2x (64x 6 – 1) Working on the parenthesis = 2x ( (8x 3 ) 2 – 1 ) Factoring as difference of squares = 2x (8x 3 + 1) (8x 3 – 1) Working on the parenthesis … = 2x ( (2x) ) ( (2x) 3 –1 ) Addition and difference of cubes = 2x(2x+1)(4x 2 –2x+1)(2x–1)(4x 2 +2x+1) Answer: 128 x 7 –162x =2x(2x+1)(4x 2 –2x+1)(2x-1)(4x 2 +2x+1)

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Ex 4: General Case Ex 4d: Factor 30 x2 x2 – 21x – 36 Taking the GCF out 30x 2 – 21x – 36 = 3(10x 2 – 7x – 12) Check if it factors as = 3(ax + p)(bx + q)q) Since ab = 10, possible values for a,b are 1&10, or 2&5 Since pq = - 12, p & q has different sign and all possible ways of getting 12 are 12(1), 6(2), 4(3). The following tables summarize the situation for pairs 1&10, 2& Each number on the last row is the adding of cross multiplication of the blue column by each one of the red columns. For example: 1 (-1) + 10 (12) = 119 … Let’s try now with p=2 & q=5. Filling the blanks on third row … Answer: 30x 2 – 21x – 36 = 3(2x - 3)(5x +4) The pair 1 & 10 doesn’t work ! ( since 7 didn’t appear on the last row ). Since we need – 7x as the middle term, so we pick p = -2 and q = 5. (Do the Check) Return to Flowchart

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Ex 5: Factor by grouping Ex 5a: Factor 3x 3 – 6x 2 + 5x –10 Splitting in two groups 3x 3 – 6x 2 + 5x –10 = 3x 3 – 6x 2 + 5x –10 Taking the GCF out from both groups = 3x 2 (x – 2) + 5 (x – 2) Taking the GCF ( x – 2 ) out (to the right) = ( 3x 2 + 5)5) Answer: 3x 3 – 6x 2 + 5x –10 = ( x – 2)( 3x 2 + 5) Ex 5b: Factor 4x 2 – 40x – 4y 2 Taking the GCF out 4x 2 – 40x – 4y 2 = 4(x 2 – 10x + 25 –y 2 ) Split expression in parenthesis into two groups = 4 ([x 2 –10x+25]– y 2 ) Factor the first group as perfect square = 4 ([ x - 5] 2 – y 2 ) Factor as difference of squares = 4 ( [x – 5 + y][x – 5 – y] ) Answer: 4x 2 – 40x –4y 2 = 4(x + y – 5)(x – y – 5) ( x – 2) Return to Flowchart

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