Factoring Quadratic Polynomials 1 copyright © 2011 Lynda Aguirre

Quadratic Polynomials copyright (c) 2011 Lynda Aguirre2 When two binomials are multiplied using the FOIL method, the answer can have 2, 3 or 4 terms. 4 terms: Multiply this using FOIL This has no like terms, so we end up with a 4-term polynomial

Quadratic Polynomials copyright (c) 2011 Lynda Aguirre3 When two binomials are multiplied using the FOIL method, the answer can have 2, 3 or 4 terms. 3 terms: Multiply this using FOIL This has like terms, so add them This polynomial now has 3-terms

Quadratic Polynomials copyright (c) 2011 Lynda Aguirre4 When two binomials are multiplied using the FOIL method, the answer can have 2, 3 or 4 terms. 2 terms: Multiply this using FOIL Combine like terms: they cancel out This polynomial now has 2-terms

copyright (c) 2011 Lynda Aguirre5 Quadratic Polynomials We just demonstrated that FOIL produces three types of polynomials Four TermsThree TermsTwo Terms None of the terms combined Middle terms combinedMiddle terms cancelled A B C D All four types of polynomials had 4-terms before cancelling or combining like terms Because of this, we can use some form of the property AD=BC to see whether the polynomial is factorable for all three types

copyright (c) 2011 Lynda Aguirre6 A B C D A-D are the coefficients of each of the four terms not including the signs Plug in the numbers to see if both sides are equal They’re equal which means this polynomial is factorable. Check for Factorability: Four Terms AD = BC AD=BC

copyright (c) 2011 Lynda Aguirre7 A B C D A-D are the coefficients of each of the four terms not including the signs Plug in the numbers to see if both sides are equal They’re equal which means this polynomial is factorable. Check for Factorability: Four Terms AD = BC AD=BC

copyright (c) 2011 Lynda Aguirre8 Check for Factorability: Four Terms AD = BC A B C D A-D are the coefficients of each of the four terms not including the signs Plug in the numbers to see if both sides are equal They’re equal which means this polynomial is factorable. AD=BC

copyright (c) 2011 Lynda Aguirre9 Check for Factorability: Three Terms AC method First(A) Middle(B) Last(C) A B and C are the coefficients of each term Because the middle term came from combining the original middle terms, we have to alter our process to find them again. A B C D Step 1: Multiply AC Step 2: Find all factors of AC Step 3: Using the sign of the last term, add or subtract the factors of AC to see if they equal the middle term Sign of the last term: - Subtract: 10-1 = 9 Subtract: 5-2 = 3 Step 4: These two factors were the original B and C Middle term= 3

copyright (c) 2011 Lynda Aguirre10 Check for Factorability: Three Terms AC method First(A) Middle(B) Last(C) A B and C are the coefficients of each term Because the middle term came from combining the original middle terms, we have to alter our process to find them again. A B C D Step 1: Multiply AC Step 2: Find all factors of AC Step 3: Using the sign of the last term, add or subtract the factors of AC to see if they equal the middle term Sign of the last term: + Add: 12+1 = 13 Add: 6+2 = 8 Step 4: These two factors were the original B and C Middle term= 8 Add: 4+3 = 7

copyright (c) 2011 Lynda Aguirre11 Check for Factorability: Two Terms A D There are two terms because the B and C cancelled each other out. (i.e. they were the same number with opposite signs) Shortcut: Step 1 Calculate AD Step 2 Find factors of AD Step 3 Look for the repeating number

copyright (c) 2011 Lynda Aguirre12 Practice: Are these polynomials factorable?