Date: Section: 4.3

Objective: Factoring trinomials of the form 2

x bx c+ + (leading coefficient 1)

These trinomials can be factored the same way we factored trinomials with a leading coefficient.

The only difference is that this time, 1.a = Follow the same instructions you did when factoring

trinomials with any other a (copied below to remind you).

Steps for Factoring a Trinomial of the Form by Grouping:

1. Always check for a GCF first! If there is a GCF, factor it out.

2. Identify a, b, and c: a is the number in front of 2 ,x b is the number in front of x, and c is

the constant (the number by itself). If one of those numbers is missing, it equals 0.

3. Multiply ⋅ .a c

4. Find two “key numbers” that multiply to your answer ( )ac and add to b.

Start by making a list of all the pairs of integers that multiply to .ac Start with 1

and work your way up until you start repeating. Then find which pair adds to b.

Think about positive and negative signs:

• Multiply to +, add to + : both are +

• Multiply to +, add to – : both are –

• Multiply to –, add to + : bigger one is +, smaller one is –

• Multiply to –, add to – : bigger one is –, smaller one is +

5. Split the middle term up: Write the polynomial as ⋅ + ⋅ +2+1st # 2nd # .ax x x c

6. Factor the resulting polynomial by grouping.

If there are no numbers that multiply to ac and add to b, the polynomial is prime.

Examples: Factor the following polynomials.

a) 2 11 30x x+ + b) 2 11 12m m+ − c) 22 40 144k k+ +

d) 2 15 56q q− + e) 2 12 45w w− − f) 25 25 30g g− + −

2ax bx c+ ++ ++ ++ +

SM 2

Have you noticed a relationship between your “key numbers” and your final answer that didn’t

happen when a wasn’t 1? When 1,a = there is a factoring shortcut!

THIS SHORTCUT DOES NOT WORK IF THERE IS A NUMBER IN FRONT OF

THE 2

x THAT CAN’T BE FACTORED OUT AS A GCF!!!! If you think you’re

going to forget when you can use it and when you can’t, stick with grouping!!!!

Shortcut (only works if 1a ==== : problem looks like 2

x bx c+ ++ ++ ++ + after factoring out GCF)!

1. Find two numbers that multiply to c and add to b.

2. The factored form of 2

x bx c+ ++ ++ ++ + is (((( )))) (((( ))))1st # 2nd # .x x+ ++ ++ ++ +

g) 2 6 9u u+ − h) 2 6 40t t− − i) 3 27 12h h h+ +

j) 2 5 6n n− − k) 2

3 10x x+ − l) 2

3 6 15x x− +

m) 2

4x − o) 2

3 27x − p) 2

144x +