Factoring trinomials in form ax2 + bx + c Algebra 1 ~ Chapter 9.4 Factoring trinomials in form ax2 + bx + c Today we will be focusing on factoring Trinomials when a is NOT equal to 1…..

First, always check to see if you can factor out a GCF first First, always check to see if you can factor out a GCF first. If the answer is yes, do so. It will make the rest of the process MUCH easier. For example, factor 2x2 + 12x + 16 Before you jump to start trying to figure out the factors, do these 3 terms have a GCF other than 1? Yes they do, we can factor out 2… 2(x2 + 6x + 8) Now we have a trinomial where a = 1 and we know how to factor easily……… 2(x + 2)(x + 4)

Example 1 – Factor 6x2 + 11x + 4 The way to factor trinomials like this is to factor by grouping…. *Want to split 11x into 2 terms so we can factor by grouping. The way we do this is to now find factors of a•c whose sum still equals b. *Since 6•4 = 24, we need to find 2 #s that multiply together to = 24 and add up to 11 Make your table and try to find them. Rewrite 6x2 + 11x + 4 as 6x2 + 3x + 8x + 4 and now factor by grouping: 3x(2x + 1) + 4(2x + 1) (3x + 4)(2x + 1)

6x2 8x 3x 4 Example 1 – AGAIN. Let’s factor the same trinomial. This time we will use the Box Method. Factor 6x2 + 11x + 4. Check your answer. Step 1: Draw a 2 x 2 table. Step 2: Plug in 6x2 in the top left and 4 in the bottom right corners. Step 3: Multiply the leading coefficient (6) and the constant term (4). Since the product is 24, we need to figure out the two #s that multiply together to = 24 and add up to 11 (the linear term)…(8 and 3) 6x2 8x 3x 4

3x 4 2x 6x2 8x 3x 4 1 Example 1 – Continued… Factor 6x2 + 11x + 4. Check your answer. Step 4: The linear terms (8 and 3) can be put in either of the remaining boxes. The order does not matter. We add the x’s because these are the “linear terms”. Step 5: Factor the GCF of the rows (across) Step 6: Factor the GCF of the columns (vertical) 3x 4 The factors of 6x2 + 11x + 4 are (2x + 1)(3x + 4) Exactly what we got on the previous 2 examples. 2x 6x2 8x 3x 4 1

Box Method of Factoring Trinomials Will only work if you factor out the GCF first. ALWAYS check your factors. Seems like a lot of work the first time through, but the pattern will come quickly. Much better than guess and check!

Example 2 - Factor the trinomial. Check your answer. 6x2 + 11x + 3 NO GCF!! 1.) Plug in 6x2 and 3 2.) 6 x 3 = 18. The 2 #s that multiply together to = 18 and add up to 11 are … 9 and 2 (linear terms) 3.) Take the GCFs of the rows and columns The factors are (3x + 1)(2x + 3) Check by FOIL – 6x2 + 9x + 2x + 3 6x2 + 11x + 3 3x 1 2x 6x2 2x  9x 3 3

Example 3 - Factor the trinomial. Check your answer. 3x2 - 2x - 8 NO GCF!! 1.) Plug in 3x2 and -8 2.) 3 x -8 = -24. The 2 #s that multiply together to = -24 and add up to -2 are … -6 and 4 (linear terms) 3.) Take the GCFs of the rows and columns (always follow the sign of the 1st term in factoring!) The factors are (3x + 4)(x - 2) Check by FOIL – 3x2 - 6x + 4x - 8 3x2 - 2x - 8 3x 4 x 3x2 4x  -6x -8 -2

Example 4 - Factor the trinomial. Check your answer. 3x2 - 16x + 16 NO GCF!! 1.) Plug in 3x2 and 16 2.) 3 x 16 = 48. The 2 #s that multiply together to = 48 and add up to -16 are … -12 and -4 3.) Take the GCFs of the rows and columns (always follow the sign of the 1st term in factoring!) The factors are (3x - 4)(x - 4) Check by FOIL – 3x2 - 12x - 4x + 16 3x2 – 16x + 16 3x -4 x 3x2 -4x  -12x 16 -4

** When the leading coefficient is negative, factor out –1 from each term before using other factoring methods.

Caution! When you factor out –1 in an early step, you must carry it through the rest of the steps and into the answer.

Example 5 - Factor the trinomial. Check your answer. -2x2 – 5x - 3 GCF is -1!! -1(2x2 + 5x + 3) 1.) Plug in 2x2 and 3 2.) 2 x 3 = 6. The 2 #s that multiply together to = 6 and add up to 5 are … 2 and 3 3.) Take the GCFs of the rows and columns The factors are -1(2x + 3)(x + 1) Check by FOIL – -1(2x2 + 2x + 3x + 3) -1(2x2 + 5x + 3) -2x2 – 5x – 3 2x 3 x 2x2 3x 2x 3 1 

Example 6 - Factor the trinomial. Check your answer. -4x2 + 19x + 5 GCF is -1!! -1(4x2 - 19x - 5) 1.) Plug in 4x2 and -5 2.) 4 x -5 = -20. The 2 #s that multiply together to = -20 and add up to -19 are … -20 and 1 3.) Take the GCFs of the rows and columns The factors are -1(4x + 1)(x - 5) Check by FOIL – -1(4x2 - 20x + 1x - 5) -1(4x2 - 19x - 5) -4x2 + 19x + 5 4x 1 x 4x2 1x -20x -5 -5 

Lesson Wrap Up Factor each trinomial. Check your answer. 1. 5x2 + 17x + 6 3. 6x2 – 23x + 7 5. –2x2 + 7x – 3 (5x + 2)(x + 3) (3x – 1)(2x – 7) -1(2x – 1)(x – 3) or (-2x + 1)(x – 3)