What is Factoring Trinomials Formula?

A Trinomial is a polynomial with three terms. Examples of Trinomial are x+y+z, x²+2x+2, x+y-1 etc. A Trinomial can be of two types. They are Perfect Square Trinomial and Non-Perfect Square Trinomial. Factoring a polynomial is nothing but writing the expression polynomial as a product of two or more expressions. A different set of steps are followed as per the given type of trinomial.

Factoring a Perfect Square Trinomial

There are two formulas to factorize the perfect square trinomial. Those are mentioned below-

a² + 2ab + b² = (a + b)²

a² – 2ab + b² = (a – b)²

One should remember that the given trinomial is a perfect square trinomial only if they are in the form of a²+2ab+b² or a²-2ab+b².

Factoring a Non-Perfect Square Trinomial

The trinomial is said to be a Non-Perfect Square Trinomial if and only if it is of the form ax²+bx+c and not a perfect square trinomial. Steps to factorize this kind of trinomial are mentioned below-

1. Determine a, b, c in a trinomial and find ac value.
2. Find two numbers whose product is ac and sum is equal to b.
3. Split the middle term in a trinomial into sum of two terms using the two numbers found in the step-2.
4. Factor by grouping.

Sample Questions

Question 1: Factorize the trinomial x² + 4x + 4.

Solution:

Given trinomial,

x²+4x+4

This can be written as-

x²+2(2)(x)+2²

It is in the form of a²+2ab+b² where a=x and b=2

So it is a perfect square trinomial and one of the formula of it can be applied in factoring.

a²+2ab+b²=(a+b)²

So, x²+4x+4=(x+2)²

(x+2)²=>(x+2)(x+2)

Question 2: Factorize the given polynomial x² – 2x + 1 using factoring trinomial formula.

Solution:

Given trinomial,

x²-2x+1

This can be written as-

x²-2(1)(x)+1²

It is in the form of a²+2ab+b² where a=x and b=1

So it is a perfect square trinomial and one of the formula of it can be applied in factoring.

a²-2ab+b²=(a-b)²

So, x²-2x+1=(x-1)²

(x-1)²=>(x-1)(x-1)

Question 3: Factorize the trinomial x² – 2x – 3.

Solution:

Given trinomial,

x²-2x-3

This cannot be written into a²+2ab+b² or a²-2ab+b². So it is not a perfect square trinomial.

So need to follow the steps to factorize a non perfect square trinomial.

Step 1: Compare given trinomial with ax²+bx+c

Where a=1,b=-2 and c=-3

ac=1×-3=-3

Step 2: Pick two numbers such that product of two numbers is equal to ac and sum of those two number is equal to b.

Let it be 1,-3

Step 3: Split the middle term into sum of two terms using above two numbers.

x²+1x-3x-3=>x(x+1)-3(x+1)

                  =(x+1)(x-3)

So, x²-2x-3=(x+1)(x-3)

Question 4: Factorize the given polynomial 3x² – 7x – 6 using factoring trinomial formula.

Solution:

Given trinomial,

3x²-7x-6

This cannot be written into a²+2ab+b² or a²-2ab+b². So it is not a perfect square trinomial.

So need to follow the steps to factorize a non perfect square trinomial.

Step 1: Compare given trinomial with ax²+bx+c

Where a=3,b=-7 and c=-6

ac=3×-6=-18

Step 2: Pick two numbers such that product of two numbers is equal to ac and sum of those two number is equal to b.

Let it be -9,2

Step 3: Split the middle term into sum of two terms using above two numbers.

3x²-7x-6=>3x²-9x+2x-6

                =3x(x-3)+2(x-3)

                =(3x+2)(x-3)

So, 3x²-7x-6=(3x+2)(x-3)

Question 5: Factorize the given trinomial 2x² – 9x + 10.

Solution:

Given trinomial,

2x²-9x+10

This cannot be written into a²+2ab+b² or a²-2ab+b². So it is not a perfect square trinomial.

So need to follow the steps to factorize a non perfect square trinomial.

Step 1: Compare given trinomial with ax²+bx+c

Where a=2,b=-9 and c=10

ac=2×10=20

Step 2: Pick two numbers such that product of two numbers is equal to ac and sum of those two number is equal to b.

Let it be -4,-5

Such that (-4)×(-5)=20 and -4+(-5)=-9

Step 3: Split the middle term into sum of two terms using above two numbers.

2x²-9x+10=>2x²-4x+(-5x)+10

                   = 2x²-4x-5x+10

                   = 2x(x-2)-5(x-2)

                   = (2x-5)(x-2)

So, 2x²-9x+10 = (2x-5)(x-2)