Factoring by splitting the middle term

A method of factoring a polynomial by splitting its middle term is called the factorization (or factorisation) by spitting the middle term.

Introduction

An expression is in the form of a trinomial $ax^2+bx+c$. In some cases, it can be factored by splitting the middle term and the process is called the factorisation (or factorization) by splitting the middle term.

Required knowledge

It is essential to have knowledge on the factoring by grouping to understand the factorization by splitting the middle term.

Steps

The quadratic expression can be factored by the following steps.

1. Write the polynomial in either ascending or descending order but it is always preferable to express the trinomial in descending order.
2. Find the product of first and last terms with their signs.
3. Try to split the middle term as either sum or difference of two terms but their product should be equal to the product of the first and last terms. Otherwise, it is not possible to factorize the trinomial.
4. Factorize the polynomial by grouping.

Example

Factorize $6x+x^2+8$

Write the expression in an order

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

$x^2+6x+8$

Find the Product of the two terms

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

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

Check the condition to split the middle term

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

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

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

Factorize the expression by grouping

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.

Problems

The list of questions with solutions to learn how to factorise an expression by splitting the middle term.

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