How to factor

Factoring, in the context of algebra, usually refers to breaking an expression (such as a polynomial) down into a product of factors that cannot be reduced further. It is the algebraic equivalent to prime factorization, where an integer is broken down into a product of prime numbers. Factoring algebraic expressions can be particularly useful for solving equations. Setting a fully factored equation equal to 0 and solving for the variables allows us to find the roots of the equation.

Factoring can be relatively simple, or fairly tricky, depending on the expression being factored. There are a number of identities for certain patterns of polynomials that can be used to factor said polynomials. The following steps can be used to factor algebraic expressions.

  1. Look for any common factors between the terms of the expression; factor the greatest common factor to reduce the expression.
  2. Check whether any identities can be used to factor the expression.
  3. Use the FOIL method backwards to factor the polynomial.
  4. If none of the above can be used, the expression cannot be factored further.

Finding common factors

To factor using common factors, determine what common factors the terms of the expression share, divide them out of the expression, and write them as a product of factors. Ideally the greatest common factor (GCF) should be used, otherwise the expression will need to be divided multiple times until it can no longer be reduced. In cases where the terms share no common factors, a different approach will need to be used, or it may not be possible to factor the expression any further.

Examples

Factor the following expressions using common factors, if possible.

1. (2x + 7):

(2x + 7) cannot be factored any further because they share no common factors.


2. (8x + 28):

(8x + 28) = 2(4x + 14) = 4(2x + 7)

In the above example, the GCF is 4. Instead of factoring 2 out of the expression twice, we could have just factored 4 instead.

3. (8x3y2 + 12x3y - 2xy):

(8x3y2 + 12x3y - 2xy) = 2xy(4x2y + 6x2 - 1)

Using factoring identities

Factoring using identities is relatively simple as long as we memorize certain polynomial patterns and their factored forms.

(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

To factor using these identities, just plug in the relevant values.

Examples

1. (x2 - 9):

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

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


2. (x2 + 4x + 4):

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

(x2 + 4x + 4) = (x + 2)(x + 2)

FOIL method

FOIL is a method for factoring that involves using the FOIL method of binomial expansion, backwards. FOIL stands for "First Outer Inner Last," which references the order in which binomials are multiplied using this method. 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. 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)

If an expression cannot be factored using any of the above methods, then the expression cannot be factored.