Factoring Trinomials of the Form
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1 Section 4 6B: Factoring Trinomials of the Form A x 2 + Bx + C where A > 1 by The AC and Factor By Grouping Method Easy Trinomials: 1 x 2 + Bx + C The last section covered the topic of factoring second degree trinomials that had 1x 2 as the first term. These trinomials looked like 1x 2 ± bx ± c. The factoring technique used in that section did not require multiple steps and many students develop their skills to the level where they can factor these trinomials in their head. For this reason we call these type of trinomials "Easy Trinomials" as in "easy to factor." Hard Trinomials: Hard Trinomials have a first term of Ax 2 where A > 1. A x 2 + Bx + C where A > 1 Examples of Hard Trinomials 6x 2 25x + 4 5x 2 16x x 2 + x 6 15x 2 x 6 Students are taught many different techniques to factor Hard Trinomials: Factoring by Grouping, The AC method, trying all the possible factors FOIL, The Box Method, and many others. A search of the internet and u tube will reveal many methods. Each of these techniques require 3 or 4 written steps. The amount of work required to factor these trinomials is why we call them "Hard Trinomials" The best technique is the one you know and are good at. If you already know a method to factor these polynomials and are good at it then continue to use that technique. If you do not know a technique then the one that follows may be considered. It is as easy as any of the other choices, and seems to be one the easiest methods for many students to learn. It seems every technique for factoring Hard Trinomials has a name. This method will be called Factoring Hard Trinomials by The AC and Factor By Grouping Method Math 120 Section 4 6B Page Eitel
2 The AC and Factor By Grouping Method to factor A x 2 + Bx + C where C is positive Example 1 Factor: 3x 2 + 7x + 2 Factor: 3x 2 + 7x + 2 3x 2 and the 2 + 6x 2 3x 2 + 7x + 2 Step 2: Find 2 terms that multiply to + 6x 2 and add to + 7x + 6x and + 1x Replace the 9x in 2x x x + 1x 2x x 2 + 6x + 1x x 2 + 6x + 1x + 2 Factor a GCF of 3x out of the first two terms and Factor a GCF of 1 out of the last two terms 3x (x + 2 ) + 1(x + 2 ) Factor a GCF of ( x + 2 ) out of the two terms ( x + 2 ) ( 3x + 1) Factor: 3x 2 + 7x + 2 (x + 2) (3x +1) Math 120 Section 4 6B Page Eitel
3 Example 2 Factor: 2x 2 9x +10 2x 2 and the x 2 2x 2 9x +10 Step 2: Find 2 terms that multiply to + 20x 2 and add to 9x 5x and 4x Replace the 9x in 2x 2 9x x 4x 2x x 2 5x 4x x 2 5x 4x + 10 Factor a GCF of x out of the first two terms and Factor a GCF of 2 out of the last two terms x (2x 5 ) 2 (2x 5 ) Factor a GCF of ( 2x 5 ) out of the two terms ( 2x 5 ) ( x 2) Factor: 2x 2 9x +10 (2x 5) (x 2) Math 120 Section 4 6B Page Eitel
4 Example 3 Factor: 12x 2 11x x 2 and the x 2 12x 2 11x + 2 Step 2: Find 2 terms that multiply to + 24x 2 and add to 11x 8x and 3x Replace the 11x in 12x 2 11x + 2 8x 3x 12x x 2 8x 3x x 2 8x 3x + 2 Factor a GCF of 4x out of the first two terms and Factor a GCF of 1 out of the last two terms 4x (3x 2 ) 1 (3x 2 ) Factor a GCF of ( 3x 2 ) out of the two terms ( 3x 2 ) ( 4x 1) Factor: 12x 2 11x + 2 (3x 2) (4x 1) Math 120 Section 4 6B Page Eitel
5 to factor Ax 2 ± Bx C where C is Negitive Example 4 Factor: 6x 2 7x 3 6x 2 and the 3 18x 2 6x 2 7x 3 Step 2: Find 2 terms that multiply to 18x 2 and subtract to 7x 9x and + 2x Replace the 7x in 6x 2 7x 3 9x + 2x 6x 2 3 6x 2 9x + 2x 3 6x 2 9x + 2x 3 Factor a GCF of 3x out of the first two terms and Factor a GCF of + 1 out of the last two terms 3x (2x 3 ) + 1 (2x 3 ) Factor a GCF of ( 3x 2 ) out of the two terms ( 2x 3 ) ( 3x + 1) Factor: 6x 2 7x 3 (2x 3) (3x +1) Math 120 Section 4 6B Page Eitel
6 to factor Ax 2 ± Bx C where C is Negitive Example 5 Factor: 12x 2 + 5x 2 12x 2 and the 2 24x 2 Step 2: Find 2 terms that multiply to 24x 2 and subtract to + 5x + 8x and 3x Replace the + 5x in 12x 2 + 5x 2 + 8x 3x 12x x 2 + 8x 3x 2 12x 2 + 8x 3x 2 Factor a GCF of 4x out of the first two terms and Factor a GCF of 1 out of the last two terms 4x (3x + 2 ) 1 (3x + 2 ) Factor a GCF of ( 3x + 2 ) out of the two terms ( 3x + 2 ) ( 4x 1) Factor: 12x 2 + 5x 2 (3x + 2) (4 x 1) Math 120 Section 4 6B Page Eitel
7 to factor Ax 2 ± Bx C where C is Negitive Example 6 Factor: 8x 2 10x 3 8x 2 and the 3 24x 2 Step 2: Find 2 terms that multiply to 24x 2 and subtract to 10x 12x and + 2x Replace the 10x in 8x 2 10x 3 12x + 2x 8x 2 3 8x 2 12x + 2x 3 8x 2 12x + 2x 3 Factor a GCF of 4x out of the first two terms and Factor a GCF of + 1 out of the last two terms 4x (2x 3 ) + 1 (2x 3 ) Factor a GCF of ( 3x 2 ) out of the two terms ( 2x 3 ) ( 4x + 1) Factor: 8x 2 10x 3 (2x 3) (4x +1) Math 120 Section 4 6B Page Eitel
8 Some Trinomials DO NOt Factor (DNF) Many trinomials cannot be factored. We write DOES NOt Factor or Prime or DNF if that is the case. Example 7 Factor: 3x 2 + 2x + 4 3x 2 and the x 2 3x 2 + 2x + 4 Step 2: Find 2 terms that multiply to + 12x 2 and add to +2 x there are no numbers that work so the hard trinomial Does Not Factor DNF Factor: 3x 2 + 2x + 4 Does not Factor DNF Prime The GCF must be taken out first before factoring the hard trinomial Math 120 Section 4 6B Page Eitel
9 To factor any polynomial completely you should factor the GCF out FIRST if the polynomial has a GCF. If the polynomial remaining inside the parentheses is a trinomial that can be factored then the complete factored answer is the product of the GCF and the factored trinomial. Example 8 Factor: 6x 2 26x + 8 Factor out the GCF of 2 ( ) 2 3x 2 13x + 4 now factor the hard trinominal 3x 2 13x + 4 3x 2 and the x 2 3x 2 13x + 4 Step 2: Find 2 terms that multiply to + 12x 2 and add to 13x 12x and 1x Replace the 13x in 3x 2 13x x 1x 3x x 2 12x 1x + 4 3x 2 12x 1x + 4 Factor a GCF of 3x out of the first two terms and Factor a GCF of 1 out of the last two terms 3x (x 4 ) 1 (x 4 ) Factor a GCF of ( x 4 ) out of the two terms ( x 4 ) ( 3x - 1) ( ) we factored out a 2 as the GCF of 6x 2 26x x 2 13x + 4 so the final factored answer is 2(x 4)(3x 1) Math 120 Section 4 6B Page Eitel
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