Definition: Factoring rewrites an expression as a product of simpler expressions, reversing multiplication; for example, x^2 + 7x + 12 factors into (x + 3)(x + 4).
What is factoring for?
A product is easier to analyze than a sum. Factoring lets you solve equations with the zero-product property, simplify fractions, and find the zeros of a function.
What is the order of steps?
First pull out the greatest common factor. Then check for a special pattern, such as a difference of squares. Then handle trinomials by finding two numbers with the right product and sum, or by grouping.
Example. Factor 4x2 – 36x + 80. The common factor is 4, giving 4(x2 – 9x + 20). Two numbers with product 20 and sum -9 are -4 and -5. So the full factorization is 4(x – 4)(x – 5).
What mistakes are common?
Stopping before the expression is fully factored, or dropping the common factor 4 once the trinomial is done. You can always check by multiplying back. Not every polynomial factors over the integers; x2 + 1 stays as it is.