Factoring Trinomials Formula

Before going to learn the factoring trinomials formulas, let us recall what is a trinomial. A trinomial is a polynomial with three terms. Factoring means writing an expression as the product of two or more expressions. A trinomial can be a perfect square or a non-perfect square. Let us learn the factoring trinomials formulas along with a few solved examples.

What Are Factoring Trinomials Formulas?

A trinomial can be a perfect square or a non-perfect square. We have two formulas to factorize a perfect square trinomial. But for factorizing a non-perfect square trinomial, we do not have any specific formula, instead, we have a process. 

• The factoring trinomials formulas of perfect square trinomials are:

  a² + 2ab + b² = (a + b)²

  a² - 2ab + b² = (a - b)²

  For applying either of these formulas, the trinomial should be one of the forms a² + 2ab + b² (or) a² - 2ab + b².

• The process of factoring a non-perfect trinomial ax² + bx + c is:

  Step 1: Find ac and identify b.

  Step 2: Find two numbers whose product is ac and whose sum is b.

  Step 3: Split the middle term as the sum of two terms using the numbers from step - 2.

  Step 4: Factor by grouping.

Let us see the applications of factoring trinomials formulas in the following section.

Examples Using Factoring Trinomials Formulas

Example 1: Factor x² - 16x + 64 using the factoring trinomials formulas.

Solution:

We can write the given trinomial as,

x² - 16x + 64 = x² - 2(x)(8) + 8²

The right side trinomial is of the form a² - 2ab + b² and hence we can apply the formula,

a² - 2ab + b² = (a - b)²

Thus, x² - 2(x)(8) + 8² = (x - 8)²

Answer: x² - 16x + 64 = (x - 8)².

Example 2: Factor the trinomial 2x² - x - 3.

Solution:

We use the factoring trinomial formula of non-perfect trinomials to factor the given trinomial.

Comparing 2x² - x - 3 with ax² + bx + c, we get a = 2, b = -1, and c = -3.

Here ac = 2(-3) = -6 and b = -1.

Two numbers whose product is -6 and whose sum is -1 are -3 and 2.

We will split the middle term -x as -3x + 2x and then we factor by grouping the terms.

2x² - x - 3

=  2x² - 3x + 2x - 3

= x (2x - 3) + 1 (2x - 3)

= (2x - 3)(x + 1)

Answer: 2x² - x - 3 = (2x - 3)(x + 1).