Factoring polynomials

Factoring polynomials involves breaking an expression down into a product of other, smaller polynomials, similar to how prime factorization breaks integers down into a product of prime factors.

There are a number of different approaches to factoring polynomials. Certain types of polynomials are relatively simple to factor, particularly when some identity or property can be used, but others can be more complicated, and require the use of methods such as the FOIL method.

Factoring out the GCF

In some cases, factoring a polynomial may be as simple as determining the greatest common factor (GCF) between the terms. To do this, look at each term in the expression to determine what shared factors they may have. Then write the new expression as a product of the GCF and the reduced terms.

Examples

Factor the following expressions.

1. 2x2 + 14x:

2x2 + 14x = 2x(x + 7)


2. 12x2 + 9x - 15:

3(4x2 + 3x - 5)


3. x + 7:

Since x and 7 share no common factors, the expression is already fully factored.

Factoring binomials

In many cases, fully factoring binomials may just require finding a GCF. Others can be factored further using identities such as difference of squares, difference of cubes, or sum of cubes:

Examples

Factor the following binomials

1. x2 - 9 (difference of squares):

(x2 - 9) = (x + 3)(x - 3)


2. x3 - 27 (difference of cubes):

x3 - 27 = (x - 3)(x2 + 3x + 9)


3. 2x3 + 54 (sum of cubes):

(2x3 + 54) = 2(x3 + 27)

2(x3 + 27) = 2(x + 3)(x2 - 3x + 9)

Factoring trinomials

Like binomials, there are a few identities that can be used to factor trinomials:

(q2 + 2qr + r2) = (q + r)(q + r)

(q2 - 2qr + r2) = (q - r)(q - r)

Trinomials that don't have the above pattern can be factored using the FOIL method.

Using the FOIL method to factor

FOIL stands for "First Outer Inner Last," which refers to a method for multiplying binomials. In the context of factoring, the FOIL method is used to help visualize the binomials that make up a polynomial. Essentially, factoring is the opposite of expanding a binomial, and can be thought of as performing the FOIL method, backwards. To factor using the FOIL method, use the following steps, and refer to the example below.

  1. Set up a product of binomials. Write 2 empty parentheses that will be filled with 2 binomials that are equivalent to the original equation.
  2. Write values for the first term in each binomial such that the product of the values is equal to the first term of the expression being factored.
  3. Find a product of two values that is equal to the third term in the expression being factored, that when added, equals the coefficient of the second term in the equation. Write each value as the second term in each binomial with the appropriate sign.

Example

Factor x2 + 3x - 28:

1. (  )(  )
2. (x  )(x  )
3. (x + 7)(x - 4)

Factoring identities

Below is a list of identities that can be used when factoring or expanding polynomials.

(q2 - r2) = (q + r)(q - r)
(q2 + 2qr + r2) = (q + r)(q + r)
(q2 - 2qr + r2) = (q - r)(q - r)
(q3 + r3) =  (q + r)(q2 - qr + r2)
(q3 - r3) =  (q - r)(q2 + qr + r2)
(q3 + 3q2r + 3qr2 + r3)  = (q + r)3
(q3 - 3q2r + 3qr2 - r3)  = (q - r)3