Warm up Factor: 1. p2 + 13p – 30 2. a2 – 12a – 45 3. x2 – 9x – 8

Lesson 5-9 Factoring Pattern for ax2 + bx + c Objective: To factor general quadratic trinomials with integral coefficients.

Trial & Error Method Example: 3x2 + 2x - 8 List the factors of the a term and the c term. a = 3 factors: 1 and 3 c = -8 factors: 2 and 4 or 1 and 8 Write down two sets of parentheses with empty spaces like this: ( x    )( x    )

Trial & Error Method 3x2 + 2x - 8 Fill the spaces in front of the x's with a pair of possible factors of the a value. There is only one possibility for our example: (3x   )(1x   ) Fill in the two spaces after the x's with a pair of factors for the constant. Let's say we choose (3x  8)(x  1).

Trial & Error Method 3x2 + 2x - 8 Decide what signs should be between the x's and the numbers. Here's a guide: If ax2 + bx + c then (x + h)(x + k) If ax2 - bx - c or ax2 + bx - c then (x - h)(x + k) If ax2 - bx + c then (x - h)(x – k) For our example 3x2 + 2x - 8 so (x - h)(x + k) (3x + 8)(x - 1)

Trial & Error Method 3x2 + 2x - 8 Test your choice by multiplying (use FOIL) the two parentheses together. Swap out your choices if necessary. In our example, let's try 2 and 4 instead of 1 and 8: (3x + 2)(x - 4) Reverse the order if necessary. Let's try moving the 2 and 4 around: (3x + 4)(x - 2)

Trial & Error Method 3x2 + 2x - 8 Double-check your signs if necessary. We're going to stick with the same order, but swap which one has the subtraction: (3x - 4)(x + 2) This finally foils to the correct trinomial.

Triple Play or “Magic” Method Example: 4x2 + 5x + 1 Multiply the a term (4 in the example) by the c term (1 in this example). 4•1 = 4 Find the two numbers whose product is this number (4) and whose sum is equal to the b term (5). 1•4 = 4   1 + 4 = 5

Triple Play or “Magic” Method Take these two numbers (which we will call h and k) and substitute them into this expression: (ax + h)(ax + k) ----------------------      a (4x + 4)(4x + 1) ----------------------      4

Triple Play or “Magic” Method Look to see which one of the two parenthesis terms in the numerator is evenly divisible by a {in this example it is (4x + 4)}. Divide this term by a and leave the other one as is. (4x + 4)(4x + 1) ----------------------      8 Answer:(x + 1)(4x + 1)

Triple Play or “Magic” Method Take the GCF (if any) out of either or both parentheses. (x + 1)(4x + 1)

Try 2b2 + 13b – 24 5y2 – 17y + 6 3k2 – 8k – 35