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Factorization

Factorization is defined as the process of breaking down a number or an expression into a product of different numbers or expressions called factors. In other words, factorization refers to breaking down large and at times complicated expressions into a product of smaller ones that are then easier to deal with. You can also think of factorization as the opposite of distribution.

For example, can be factored into by extracting the common factor of 3x.

Factorization can be done in three ways:

  • by taking the difference of two squares
  • by grouping and
  • by breaking up the expression into perfect trinomials.

Factorization by Taking Difference of Two Squares

The difference of two squares is also known as the difference of perfect squares and refers to the mathematical identity

The above identity is true regardless of the the coefficients of a and b, given that the coefficients are the same for a and b.

This is because expanding (a + b)(a - b) results in similar terms that cancel each other out, i.e.

ab and ba are the same and thus can be added or subtracted, and in this case their difference is zero, which leaves us with

Certain polynomials can be factored using the difference of two squares, but these polynomials must have squares in them. For example

can be factored using the difference of two squares into

Similarly

can be factored using the same method into

This can be further factored using the same method into

By equating the polynomial to zero, the roots of the polynomial can be easily found from the factors.

Factorization by Grouping

Factorization by grouping involves separating the given expression into smaller groups and then factoring those groups independently. The tricky part about this becomes choosing which expressions to group together. It wouldn't make much sense to group expressions with odd coefficients with those with even coefficients but there is no consensus on how to pick the different groups. It's up to you to decide what makes sense, and the more practice you have with grouping, the more obvious it becomes.

For example, factor

solution

The first step is to pick which expressions to group together. In this case it should be obvious from the coefficients.

Observe that the operator between the different sets of parentheses is an addition operator (+). This is important to remember because only sum / difference operators are allowed.

Factoring the separate groups results in

The above is done by extracting the common factors to each set of parentheses.

Now observe that (x + 3) is a common factor to both expressions and can be factored out as:

Its important to remember that you should NEVER divide through by anything since you might lose some factors by doing so.

Factorization into Perfect Square Trinomials

Perfect square trinomials refer to quadratic polynomials which can be factored into identical binomial expressions. These quadratic polynomials are of the standard form:

and

For more on factoring quadratic polynomials, refer to section on quadratic equations.

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