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Section 6.5 A General Factoring Strategy Difference of Two Squares: OTE: Sum of Two Squares, , is not factorable Sum and Differences of Two Cubes: Perfect Square Trinomials: 2 2 Strategy for Factoring Polynomials Step 1: Factor out GCF, if possible Step 2: Determine the number of terms in the polynomial Step 3: Determine a factoring technique as follows: If the given polynomial is a Binomial, factoring by one of the following 1. 2. 3. If the given polynomial is a Trinomial, factoring by ac-method If the given polynomial has 4 or more terms, factoring by Grouping Exercises (Solution 1) Step 1: Factor out GCF. GCF = 2x 2 8 2 4 Since 4 is the combination of two linear, the polynomial is factored completely. Factoring is done! The answer is 2 4 or 2 4 Cheon-Sig Lee www.coastalbend.edu/lee Page 1 Section 6.5 A General Factoring Strategy (Solution 2) Step 1: Factor out GCF. GCF = 1 Step 2: Determine the number of terms: Two terms Step 3: Determine a factoring technique Since two terms, power 2, and difference, for factoring we are using Factoring by Step 1: Rewrite in terms of perfect squares 4 25 2 5 2 5 Step 2: Find a and b, then factor out with ⟹ 2 , 5 Two linear combination, so factoring is done (Solution 3) Step 1: Factor out GCF. GCF = 1 Step 2: Determine the number of terms: Two terms Step 3: Determine a factoring technique Since two terms, power 3, and difference, we are using for factoring Factoring by Step 1: Rewrite in terms of perfect squares 216 1 6 1 6 1 Step 2: Find a and b, then factor out with 6 , 1 1 ⟹ 6 1 36 6 1 (Solution 4) 2 2 is four terms, so By Grouping Factoring by Grouping Step 1: Group first two terms and last two terms 2 2 Step 2: Factor out GCF from each group GCF for the first group is 2 GCF for the second group is x 2 2 2 2 Cheon-Sig Lee www.coastalbend.edu/lee Page 2 Section 6.5 A General Factoring Strategy (Solution 5) Ignore FOIL multiplication Step 1: Factor out GCF. GCF is 1, so DONE! Step 2: Find a, b, c 8, 22, 15 Step 3: Find ac 8 ∙ 15 120 Step 4: Find positive factors of ac whose products is ac and whose sum is ‘b’ Positive factors of 40: 1 120 , 2 60 , 3 40 , 4 30 5 24 , 6 20 , 8 15 , 10 12 Since ac is positive, there are two case; Case 1: Both factors are positive Case 2: Both factors are negative Since ‘b’ is negative, both factors are negative Thus, possible factors are 1 120 , 2 60 , 3 40 4 30 , 5 24 , 6 20 8 15 , Step 5: Rewrite the middle term, bx, using the two numbers found in step 4 as coefficients. Then factoring by grouping. 8 22 15 8 15 8 15 2 3 2 3 Ignore FOIL multiplication (Solution 6) Step 1: Factor out GCF. GCF = 3x 3 12 4 4 2 ⟹ Sum of Two Squares The sum of two squares is not factorable So, factoring is done! The answer is 3 4 Cheon-Sig Lee www.coastalbend.edu/lee Page 3 Section 6.5 A General Factoring Strategy (Solution 7) Step 1: Factor out GCF. GCF = 5x 10 5 2 1 Since 5 2 1 is the combination of two linear, the polynomial is factored completely. Factoring is done! The answer is 5 2 1 (Solution 8) Step 1: Factor out GCF. GCF is 1, so DONE! Step 2: Find a, b, c 1, 5, 25 Step 3: Find ac 1 ∙ 25 25 Step 4: Find positive factors of ac whose products is ac and whose sum is ‘b’ Positive factors of 25: 1 25 , 5 5 Since ac is positive, there are two case; Case 1: Both factors are positive Case 2: Both factors are negative Since ‘b’ is negative, both factors are negative Thus, possible factors are 1 25 , 5 5 1 25 26 5 5 10 No combination gives the middle term – 5. Thus, the polynomial is prime. (Solution 9) Step 1: Factor out GCF. GCF = 3x 3 27 9 Step 2: Determine the number of terms: 9 has two terms Step 3: Determine a factoring technique Since two terms, power 2, and difference, we are using for factoring Factoring by Step 1: Rewrite in terms of perfect squares 3 9 3 Step 2: Find a and b, then factor out with ⟹ , 3 3 3 Three linear combination, so factoring is done Cheon-Sig Lee www.coastalbend.edu/lee Page 4 Section 6.5 A General Factoring Strategy (Solution 10) Step 1: Factor out GCF. GCF = 1 Step 2: Determine the number of terms: Two terms Step 3: Determine a factoring technique Since two terms, power 2, and sum, it is prime 36 6 ⟹ Remember! Sum of Two Squares is not factorable Cheon-Sig Lee www.coastalbend.edu/lee Page 5