What are complex trinomials and how to factor them?

 

How are they different from simple trinomials?

• Complex trinomials have a co-efficient in front of x^2 that is greater than 1.

• For example, 3x^2 +13x+12 is a complex trinomials because the co-efficient in front of x^2 is 3 (greater than 1).

 

It is in the form of ax^2+bx+c

The "a" value is greater than 1.

 

How to factor them?

Using the guess and check method!

 

• The factored form is written as (____) (____).

• They follow the same rules as the simple trinomials where the product of the two numbers is the "c" value, and their sum is the "b" value.

 

 

 In simple trinomials we had (x__) (x__) as the two variables, to get x^2.

However, in complex trinomials the co-effiecient is greater than 1, so we have to guess the first two terms.

• For example, if the expression is 3x^2+13x+12, the possible value is (x___) (3x___). 

• These terms need to be factors of 3, so in this case they are 1 and 3.

 

What if it was 12x^2+13x+12?

• The factors of 12 are 1,2,3,4,6, and 12.

• So possible terms are (x___) (12x___), (2x___) (6x___), (3x___) (4x___).

• x multiplied by 12x is 12x^2.

• 2x multiplied by 6x is 12x^2.

• 3x multiplied by 4x is 12x^2.

 

Then we need to find the other two numbers that are factors of "c", and their product is equal to "c" while their sum is equal to "b".

 

Let's try an example!

3x^2+13x+12

• First find the factors of 3!

• Factors of 3 are 1 and 3, so the only possible terms are (x___) (3x___).

• Now find the factors of 12!

• Factors of 12 are 1,2,3,4,6, and 12.

• Wait, none of them add up to 13!

• Do not forget that we have a 3x, so we need to multiple one of them by 3.

• Let's try 2 and 6!

• (x+2) (3x+6).

• To check, let's expand.

• x multiplied by 3x is 3x^2.

• x multiplied by 6 is 6x.

• 2 multiplied by 3x is 6x.

• 2 multplied by 6 is 12.

• Collect like terms which are 6x+6x=12x.

• So we get 3x^2+12x+12.

• Does this match the original equation?

• Nope!

 

Then what is the answer?

• The answer is 3 and 4.

• Let's see how.

• (x+3) (3x+4).

• x multiplied by 3x is 3x^2.

• x multiplied by 4 is 4x.

• 3 multiplied by 3x is 9x.

• 3 multiplied by 4 is 12.

• Let's collect like terms which are 4x+9x=13x.

• We get 3x^2+13x+12.

• So we are right!

 

How do we find this?

Well there is no real way to find it, however, you need to do it mentally and check if (x multiplied by 4)+(3x multiplied by 3) is equal to 13x.

 

Want to try another one?

Sure!

Let's try 6x^2+11x+4

• Find factors of 6.

• Factors of 6 are 1,2,3 and 6.

• Since 11 is not a big number, let's try 3 and 2 since they have a smaller difference (difference of 1, compared to difference of 5 if we use 1 and 6).

• (2x___) (3x___).

• Now time to find factors of 4.

• Factors of 4 are 1,2, and 4.

• Try mentally which ones equal to 11x when added.

• Let's try 4 and 1.

• (2x+1) (3x+4)

• 2x multiplied by 3x is 6x^2.

• 2x multiplied by 4 is 8x.

• 1 mutiplied by 3x is 3x.

• 1 multiplied by 4 is 4.

• Collect like terms which are 8x+3x=11x.

• We get 6x^2+11x+4.

• Therefore, we are correct!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Practice these...

1) 6x^2+11x+3

2) 2y^2+11y+15

3) 5m^2+14m+8

4) 3d^2+17d+10

5) 6m^2-17m+5

 

 

Answers are...

 

 

 

 

1) (3x+1) (2x+3)

2) (y+3) (2y+5)

3) (m+2) (5m+4)

4) (d+5) (3d+2)

5) (2m-5) (3m-1)

 

There are several ways to factor complex trinomials!

If you want to try another way then utilize this website-https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/factoring-by-grouping/v/factoring-trinomials-with-a-non-1-leading-coefficient-by-grouping