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.