Math Doubts

Factoring by splitting the middle term

A method of factoring a trinomial by splitting the middle term of an expression or polynomial is called the factorization (or factorisation) by splitting the middle term.

Introduction

An expression often appears with three terms in mathematics and it can be factored mathematically in some cases but the polynomial should have the following two properties.

  1. It should be in the form $ax^2+bx+c$
  2. Possibility of splitting the middle term as two terms in such a way that their product is equal to the product of first and last terms of the trinomial.

Required knowledge

You must study the following concept to understand the factorization of expressions or polynomials by splitting the middle term of trinomials.

  1. Factoring by Grouping

Steps

The factoring of an expression that contains three terms can be done by the following steps.

  1. Write the polynomial in descending order.
  2. Find the product of first and last terms with their signs.
  3. Try to split the middle term of the trinomial as sum or difference of two terms but their product should be equal to the product of the first and last terms of the expression. If possible, then continue the process of the factorization. Otherwise, it’s not possible to factorise the trinomial in this method.
  4. Now, arrange the four terms as two groups and factorize the polynomial by grouping.

Example

Factorize the trinomial $x^2+6x+8$.

Step – 1

The given algebraic expression is already in descending order. So, need to do anything with the given polynomial.

$x^2+6x+8$

Step – 2

Find the product of first and last terms of the expression.

$x^2 \times 8 \,=\, 8x^2$

Step – 3

Try to split the middle term $6x$ as two terms but their product should be equal to $8x^2$.

$2x+4x = 6x$ and $(2x)(4x) = 8x^2$.

$= \,\,\,$ $x^2+2x+4x+8$

Step – 4

Now, group the terms to factorize the algebraic expression by grouping.

$= \,\,\,$ $(x^2+2x)+(4x+8)$

$= \,\,\,$ $x(x+2)+4(x+2)$

$= \,\,\,$ $(x+2)(x+4)$

Therefore, the given trinomial $x^2+6x+8$ is factored as $(x+2)(x+4)$ by splitting the middle of the given expression.

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