HFCC Math Lab Beginning Algebra -19. In this handout we will discuss one method of factoring a general trinomial, that is an

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1 HFCC Math Lab Beginning Algebra -19 FACTORING TRINOMIALS a + b+ c ( a In this handout we will discuss one method of factoring a general trinomial, that is an epression of the form a + b+ c where a, b, and c are constant, and usually integers. In particular, we will be concerned with the case where the leading coefficient is not one ( a. When the leading coefficient is one ( a=, a shorter two-step method can be used to factor the trinomial. REMEMBER that in every case you should FIRST check for a COMMON FACTOR before proceeding with any other method of factoring. The procedure to factor a trinomial of the form a + b+ c ( a is as follows: Find the product of a and c, ac. ) Find two integersm and n such that the product of m and n is ( ac ) and the sum of m and n is the coefficient of the middle term ( b ). That is: the product: ac= m n and the sum: b= m+ n. 3) Rewrite the trinomial, using the numbers you found in step to break the middle term, that is, rewrite a + b+ c as a + m+ n+ c. Note that since b= m+ n, so also b= m+ n, and therefore the rewritten epression a + m+ n+ c is equivalent to the original trinomial a + b+ c. ) Factor out the Greatest Common Factor (GCF) from the first two terms a + m, and the GCF or GCF from the second two terms n+ c Factor out the resulting common binomial factor. The above steps will become clearer as you follow through some eamples. Factor 1 13 completely: FIRST: check for COMMON FACTOR among the coefficients or the variables. In this case the COMMON FACTOR is 1. Factoring 1 out will not simplify the problem. NOTE: a=+ 1, b= 13, and c= Step 1: Find the product: ac= ( + ( ) = 8 Revised 03/09 1

2 Step : Find m, n where m n= 8 and m+ n= 13, here m=+ 3 and n= 1 Step 3: Rewrite the trinomial using the two integers from step to break up the 1 13 = Step : = 3( + ( Step 5: =( ( 3 ) + Split the middle term + Factor the GCF from the first two terms and + Factor out the resulting common binomial factor ( =( ( 3 ) +. 3 ) Factor completely: FIRST: check for a COMMON FACTOR among the coefficients or the variables. In this case the COMMON FACTOR = ; Factoring out 8 9+ this factor we get: ( ) We will now use the procedure given above to factor8 9+. We must remember to include the factor ( ) in our final answer. NOTE: a=+ 8, b= 9, and c=+ Step 1: Find the product: ac= ( + 8)( + ) =+ 8 Step : Find m, n where m n=+ 8 and m+ n= 9, here m= 1 and n= 8 Step 3: Rewrite the trinomial using the two integers from step to break up the = 8 8+ Split the middle term Step : ( 8 ( 8 = Factor the GCF from the first two terms and Revised 03/09

3 Step 5: ( 8 ( ) = Factor out the resulting common binomial factor ( = ( 8 ( ). 3) Factor completely: FIRST: check for COMMON FACTOR among the coefficients or the variables. In this case the COMMON FACTOR is 1. Factoring 1 out will not simplify the problem. NOTE: a=+, b=+ 19, and c=+ Step 1: Find the product: ac= ( )( ) = 0 Step : Find m, n where m n= 0 and m+ n=+ 19, here m=+ and n=+ 15 Step 3: Rewrite the trinomial using the two integers from step to break up the = Split the middle term Step : ( 3 ) 5( 3 ) = Factor the GCF from the first two terms and Step 5: ( 3 )( = + + Factor out the resulting common binomial factor ( 3+ ) = ( 3 )( = + +. ) Factor 19 + completely: ( Compare this to eample 3) FIRST: check for COMMON FACTOR among the coefficients or the variables. In this case the COMMON FACTOR is 1. Factoring 1 out will not simplify the problem. NOTE: a=+, b= 19, and c=+ Revised 03/09 3

4 Step 1: Find the product: ac= ( )( ) = 0 Step : Find m, n where m n= 0 and m+ n= 19, here m=, n= 15 Step 3: Rewrite the trinomial using the two integers from step to break up the 19 + = 15+ Split the middle term Step : ( 3 ) 5( 3 ) = Factor the GCF from the first two terms and Step 5: ( 3 )( = Factor out the resulting common binomial factor ( 3 ) 19 + ( 3 )( =. Factor completely: Before you start rewrite in descending order FIRST: check for a COMMON FACTOR among the coefficients or the variables. In this case the COMMON FACTOR = 1 ; Factoring out this factor we get: ( ) We will now use the procedure given above to factor Generally it is easier to factor a trinomial if the leading coefficient a is positive We must remember to include the factor ( in our final answer. NOTE: a=+ 3, b= 11, and c= Step 1: Find the product: ac= ( 3)( ) = 1 and Step : Find m, n where m n= 1 and m+ n= 11, here m= 1 and n=+ 1 Step 3: Rewrite the trinomial using the two integers from step to break up the 3 11 Revised 03/09

5 = Split the middle term Step : 3( ) 1( ) = + Factor the GCF from the first two terms and Step 5: ( )( 3 = + Factor out the resulting common binomial factor ( ) = = ( ) ( ( )( 3 = +. ) Factor 3 + 7y y completely: FIRST: check for COMMON FACTOR among the coefficients or the variables. In this case the COMMON FACTOR is 1. Factoring 1 out will not simplify the problem. NOTE: a=+ 3, b=+ 7, and c= Step 1: Find the product: ac= ( 3)( ) = 18 Step : Find m, n where m n= 18 and m+ n=+ 7, here m=+ 9 and n= Step 3: Rewrite the trinomial using the two integers from step to break up the 3 + 7y y = 3 + 9y y y Split the middle term Step : 3( 3y) y( 3y) = + + Factor the GCF from the first two terms and Step 5: ( 3y)( 3 y) = + Factor out the resulting common binomial factor ( + 3y) 3 7y y + ( 3y)( 3 y) = +. Revised 03/09 5

6 7) Factor + completely: ac= = 0 9 Step 1: Find the product: ( )( ) Step : Find m, n where m n= 0 and m+ n=+ 9 Checking carefully: None of the m n= 0 options will result in m+ n=+ 9 ( 1 + 0, +, + 5, + 1 0, +, + Therefore: The quadratic trinomial + 9 cannot be factored using integer coefficients. EXERCISES: Factor out the following trinomials completely: ) 3) ) ) 7) 8) y 37y+ 7 a a ) 3r rs+ 7s 3 1) y+ y 1) 3y y 3 17) 15a c+ ac 5c 18) 8 y + y + y 3 19) + 11p p 0) 1 y 5y 9) 5+ 7 ) + 13y+ y 1 5a ab b c cd+ d + 30w 50w ) t 7t+ 3) 18a b ab + b 3 ) y yz+ 3z Revised 03/09

7 Answers with solutions for most problems.: m n= and m n = + ( ) 1( ) ( )( = + = + + = ( m=, n=+ ) m n= and m n 3 3 = + 1 ( ) 1( ) ( )( = + = + + = ( m=, n=+ 3) m n= 1 and m n = ( 3 ( 3 ( 3 ( ) = + + = + + = ( m=+ 1, n= ) ( 3)( m n= 3 and m n = + 9 ( 3 ) 3( 3 ) ( 3 )( 3) = = ) ( 3 )( ) + = ( m=, n= 9) Revised 03/09 7

8 7) m n= 70 and m n y y + = + y y 35y 7 y( 5y 7( 5y ( 5y ( y 7) = = 8) m n= and m n 3 3 = + 1 ( ) 1( ) ( 5a ( a ) = + = + 9) m n= 1 and m+ n= 5 + = ( m=, n= 3 + = ( m=, n=+ No integers work not factorable using integers. ) = ( + y)( + y) 1 m n= 0 and m n 1 5a ab b = 5a + ab 5ab b a( 5a b) b( 5a b) ( 5a b)( a b) = + + = + + = ( m=+, n= No integers work not factorable using integers. 13) m n= 1 and m n 3r rs + 7s = 3r 1rs 1rs+ 7s r( 3r s) 7s( 3r s) ( 3r s)( r 7s) = = 1) = ( 3 ( + ) + = ( m= 1, n= ) Revised 03/09 8

9 y+ y ( y y ) = = 3 + 1y+ 1y+ y ( y) y( y) ( y)( y) = = Note: m n=+ 1 and m+ n=+ 13 ( m=+ 1, n=+ 1) = ( y+ ( 8y 9) 17) a ac c ( ) = 5c 3a + a 5 = + 5c 3a 3a 5a 5 ( ) ( ) ( )( a ) = 5c 3a a a1 = 5c a Note: m n= 15 and m+ n=+ ( m= 3, n=+ 18) = y ( + ( + NOTE: You can get additional instruction and practice by going to the following websites: This website gives eplanations and eamples. Be sure to click on NEXT at the bottom of each page for continued instruction. ol_alg_tut7_factor.htm This website gives step-by-step instruction along with practice problems and videos for some eamples. Revised 03/09 9