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**Flowchart to factor Factor out the Great Common Factor**

Ex 1 Can be the expression written as A2 – B2 or A+ B3 or A3 – B3 ? YES Ex 2 Factor Two terms? YES NO END NO Can the expression be written as ax2+bx+c? YES YES perfect square? Ex 3 Three Terms? YES NO NO product of two binomials? YES END NO Ex 4 NO END Can you factorize the expression by grouping ? Four or More Terms? YES YES Ex 5 NO END

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**Ex 1: Greatest Common Factor (GCF)**

Ex 1a: Factor 6ab+8ac+4a Look for the GCF ab+8ac+4a = 2 ·3 ab + 2 · 4ac + 2 ·2a Take out the GCF = 2a ( ) Working on the parenthesis = 2a ( 3 b c ) Answer: 6ab+8ac+4a = 2a ( 3b + 4c + 2) Ex 1b: Factor 5x2y(2a–3) –15xy(2a-3) Look for the GCF 5x2y(2a–3)–15xy(2a-3) = 5xxy(2a-3)– 3•5xy(2a-3) Take out the GCF = 5xy(2a-3)( ) Working on the parenthesis = 5xy(2a-3)(x ) Answer: 5x2y(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 3x3 – 12xy2 Taking the GCF out x3 – 12xy2 = 3x ( x2 – 4y2 ) Factoring as difference of squares = 3x (x+2y)(x -2y) Answer: 3x3 – 12xy2 = 3x (x+2y(x -2y) Ex 2b: Factor 16x4 + 54x2 Taking the GCF out x5 + 54x2 = 2x2 (8x ) Working on the parenthesis = 2x2( [2x]3 + [3]3 ) Factoring as difference of cubes = 2x2 (2x+ 3)[ (2x)2 –(2x)3 +(3)2 ] Simplifying = 2x2 (2x+ 3 )(4x2 - 6x + 9) Answer : 16x5 – 54x2 = 2x2(2x + 3)(4x2 – 6x + 9) Return to Flowchart

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**Ex 3: Factoring as Perfect Square**

Ex 3a: Factor 63x4 – 210x x2 Taking GCF out x4 – 210x x2 = 7x2 (9x2 – 30x + 25) Check for perfect square ( )2 Take square root first and last terms = 7x2 (3x – )2 Middle term’s check (3x)5 = 30x OK! Answer: 63x4 – 30x3 + 25x2 = 7x2 (3x – 5 )2 Ex 3b: Factor 3x3y2 +18x2y +27x Taking the GCF out x3y2 + 18x2y +27x = 3x (x2y2 + 6xy + 9) Check for perfect square ( )2 Take square root first and last terms = 3x ( xy )2 Middle term’s check (xy)3 = 6xy OK! Answer: 3x3y2 +18x2y + 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 3x3 – 21x2 + 30x Taking GCF out x3 – 21x2 + 30x = 3x (x2 – 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 = both should be negative. Possibilities for a, b are 1,10 or 2,5. Let’s check x x - x - 10x -11x x - 2 x - 5 -2x -5x -7x NO! OK! So, a = -2 and b = x3 – 21x2 + 30x = 3x (x -2)(x –5) Answer: 3x3 – 21x2 + 30x = 3x (x - 2)(x - 5) Return to Flowchart

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**Ex 4: Simple Case Continued …**

Ex 4b: Factor 4x5 – 20x3 + 16x Taking the GCF out x5 – 20x3 + 16x = 4x (x4 – 5x2 + 4 ) Trinomial is not a perfect square ! Factoring as product of two binomials = 4x (x2 – 1) (x2 – 4) Factoring as difference of squares =4x (x+1)(x–1)(x+2)(x–2) Answer: 4x5– 20x3+16x = 4x (x+1)(x–1)(x+2)(x–2) Ex 4c: Factor 128 x7 – 2x Taking the GCF out … x7 – 2x = 2x (64x6 – 1) Working on the parenthesis = 2x ( (8x3 )2 – 1 ) Factoring as difference of squares = 2x (8x3 + 1) (8x3 – 1) Working on the parenthesis … = 2x ( (2x)3+ 1 ) ( (2x)3 –1 ) Addition and difference of cubes =2x(2x+1)(4x2 –2x+1)(2x–1)(4x2+2x+1) Answer: 128 x7–162x =2x(2x+1)(4x2–2x+1)(2x-1)(4x2+2x+1)

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**Ex 4: General Case 7 Ex 4d: Factor 30 x2 – 21x – 36**

Taking the GCF out x2 – 21x – 36 = 3(10x2 – 7x – 12) Check if it factors as = 3(ax + p)(bx + 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&5 119 - 2 58 14 26 37 1 12 6 2 3 4 10 - 1 -12 - 2 - 6 - 4 - 3 2 12 1 6 3 4 5 - 1 -12 - 2 - 6 - 4 - 3 58 -19 26 14 2 12 1 6 3 4 5 - 1 -12 - 2 - 6 - 4 - 3 7 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 … The pair 1 & 10 doesn’t work ! ( since 7 didn’t appear on the last row). Let’s try now with p=2 & q=5. Filling the blanks on third row … Since we need – 7x as the middle term, so we pick p = -2 and q = 5. (Do the Check) Answer: 30x2 – 21x – 36 = 3(2x - 3)(5x +4) Return to Flowchart

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**Ex 5: Factor by grouping Ex 5a: Factor 3x3 – 6x2 + 5x –10**

Splitting in two groups x3 – 6x2 + 5x –10 = 3x3 – 6x x –10 Taking the GCF out from both groups = 3x2 (x – 2) + 5 (x – 2) Taking the GCF ( x – 2 ) out (to the right) = ( 3x2 + 5) Answer: 3x3 – 6x2 + 5x –10 = ( x – 2)( 3x2 + 5) Ex 5b: Factor 4x2 – 40x – 4y2 Taking the GCF out x2 – 40x – 4y2 = 4(x2 – 10x + 25 –y2) Split expression in parenthesis into two groups = 4 ([x2–10x+25]– y2 ) Factor the first group as perfect square = 4 ([ x ]2 – y2 ) Factor as difference of squares ( x – 2) = 4 ( [x – 5 + y][x – 5 – y] ) Answer: 4x2 – 40x –4y2 = 4(x + y – 5)(x – y – 5) Return to Flowchart

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