# What is Factoring Trinomials Formula?

A Trinomial is a polynomial with three terms. Examples of Trinomial are x+y+z, x^{2}+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-

a

^{2 }+ 2ab + b^{2 }= (a + b)^{2}a

^{2 }– 2ab + b^{2 }= (a – b)^{2}

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

### 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 ax^{2}+bx+c and not a perfect square trinomial. Steps to factorize this kind of trinomial are mentioned below-

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

### Sample Questions

**Question 1: Factorize the trinomial x ^{2 }+ 4x + 4.**

**Solution:**

Given trinomial,

x

^{2}+4x+4This can be written as-

x

^{2}+2(2)(x)+2^{2}It is in the form of a

^{2}+2ab+b^{2}where a=x and b=2So it is a perfect square trinomial and one of the formula of it can be applied in factoring.

a

^{2}+2ab+b^{2}=(a+b)^{2}

So, x^{2}+4x+4=(x+2)^{2}(x+2)

^{2}=>(x+2)(x+2)

**Question 2: Factorize the given polynomial x ^{2 }– 2x + 1 using factoring trinomial formula.**

**Solution:**

Given trinomial,

x

^{2}-2x+1This can be written as-

x

^{2}-2(1)(x)+1^{2}It is in the form of a

^{2}+2ab+b^{2}where a=x and b=1So it is a perfect square trinomial and one of the formula of it can be applied in factoring.

a

^{2}-2ab+b^{2}=(a-b)^{2}

So, x^{2}-2x+1=(x-1)^{2}(x-1)

^{2}=>(x-1)(x-1)

**Question 3: Factorize the trinomial x ^{2 }– 2x – 3.**

**Solution:**

Given trinomial,

x

^{2}-2x-3This cannot be written into a

^{2}+2ab+b^{2}or a^{2}-2ab+b^{2}. 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 ax^{2}+bx+cWhere a=1,b=-2 and c=-3

ac=1Ã—-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.x

^{2}+1x-3x-3=>x(x+1)-3(x+1)=(x+1)(x-3)

So, x^{2}-2x-3=(x+1)(x-3)

**Question 4: Factorize the given polynomial 3x ^{2 }– 7x – 6 using factoring trinomial formula.**

**Solution:**

Given trinomial,

3x

^{2}-7x-6This cannot be written into a

^{2}+2ab+b^{2}or a^{2}-2ab+b^{2}. 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 ax^{2}+bx+cWhere a=3,b=-7 and c=-6

ac=3Ã—-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.3x

^{2}-7x-6=>3x^{2}-9x+2x-6=3x(x-3)+2(x-3)

=(3x+2)(x-3)

So, 3x^{2}-7x-6=(3x+2)(x-3)

**Question 5: Factorize the given trinomial 2x ^{2 }– 9x + 10.**

**Solution:**

Given trinomial,

2x

^{2}-9x+10This cannot be written into a

^{2}+2ab+b^{2}or a^{2}-2ab+b^{2}. 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 ax^{2}+bx+cWhere a=2,b=-9 and c=10

ac=2Ã—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)Ã—(-5)=20 and -4+(-5)=-9

Step 3:Split the middle term into sum of two terms using above two numbers.2x

^{2}-9x+10=>2x^{2}-4x+(-5x)+10= 2x

^{2}-4x-5x+10= 2x(x-2)-5(x-2)

= (2x-5)(x-2)

So, 2x^{2}-9x+10 = (2x-5)(x-2)

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