1.3 Factoring Methods

Factoring Methods

In this section, we examine three steps in factoring a polynomial:

1. Factor out GCF
2. Factor difference of two squares
3. Factor a polynomial of the form $x^2 + bx + c$

These factoring steps are often used as part of the solution method for solving polynomial equations.

Factoring a polynomial means to rewrite the expression as a multiplication.

If we were to multiply the expression “$2x$” times the expression “$(3x + 7)$” we can multiply using the distributive property:

$2x(3x + 7)$

$= 2x(3x) + 2x(7)$

$=6x^2 + 14x$

Factoring is the inverse operation to multiplication so to factor this expression $6x^2 + 14x$ means to rewrite as the original multiplication.

Thus the factoring process would look like:
$6x^2 + 14x = 2x(3x + 7)$

Factoring the Greatest Common Factor (GCF)

The expression “$2x$” is called the Greatest Common Factor (GCF).

The first step in any factoring attempt is to first determine if a Greatest Common Factor is present.  To identify the Greatest common factor, break this down into two steps:

1. Examine the coefficients in the polynomial and identify the greatest integer which evenly divides into all of the coefficients.
2. Examine the variable expressions and identify the greatest expression which evenly divides into the variable expressions in the polynomial. Note: The variable part of the greatest common factor always contains the smallest power of a variable that appears in all terms of the polynomial.

Example:
Factor $27x^3 + 36x^2$

Solution:

1. Examine the coefficients in the polynomial and identify the greatest integer which evenly divides into all of the coefficients.  Note that “9” is the greatest integer that evenly divides into both 27 and 36.
2. Examine the variable expressions and identify the greatest expression which evenly divides into the variable expressions in the polynomial. Note: The variable part of the greatest common factor always contains the smallest power of a variable that appears in all terms of the polynomial. Note that $x^2$ is the variable expression with the smallest power.

Thus the GCF is $9x^2$.

This expression is then factored out of the original polynomial as follows:
$9x^2(3x + 4)$

Factoring the Difference of Two Squares

When factoring a polynomial with two terms check if the polynomial represents the difference of two squared quantities. For example the expression $x^2 - 25$ is the difference of two squared quantities since x is being squared in the first expression and 5 is being squared in the second expression.

We can rewrite $x^2 - 25$ as $(x)^2 - (5)^2$ to emphasize the difference of the two squared quantities.

The difference of two squares if factored according to the following template:

$A^2 - B^2 = (A + B)(A - B)$

The example of $x^2 - 25$, which is rewritten as $(x)^2 - (5)^2$ can thus be factored as:
$(x + 5)(x - 5)$

Example:
Factor $9x^2 - 144y^2$

Solution:
Rewrite the expression to show two squared quantities:

$9x^2 - 144y2 = (3x)^2 - (12y)^2$

Apply the template for factoring the difference of two squares:

$(3x)^2 - (12y)^2$

$=(3x + 12y)(3x - 12y)$

Factoring Trinomials of the form $x^2 + bx + c$

When factoring a trinomial whose leading coefficient is 1, we attempt to factor using the product of two binomials.

Example:
Factor $x^2 + 7x + 12$

Solution:
We form the multiplication of two binomials as follows:

$(x + \_ )(x + \_)$

We must now find two integers whose product is the constant 12 and whose sum is the coefficient of the x-term which is 7.

The two integers are then 3 and 4, so we populate the two binomials as:

$(x + 3)(x + 4)$.

Example:
Factor $x^2 - 9x + 20$

Solution:
We form the multiplication of two binomials as follows:

$(x + \_ )(x + \_)$

We must now find two integers whose product is the constant 20 and whose sum is the coefficient of the x-term which is -9.

The two integers are then -4 and -5, so we populate the two binomials as:

$(x - 4)(x - 5)$.

Example:
Factor $x^2 - 7x - 30$

Solution:
We form the multiplication of two binomials as follows:

$(x + \_ )(x + \_)$

We must now find two integers whose product is the constant -30 and whose sum is the coefficient of the x-term which is -7.

The two integers are then 3 and -10, so we populate the two binomials as:

$(x + 3)(x - 10)$.