A polynomial is an algebraic expression with more than one term. In this case, the polynomial will have four terms, which will be broken down to monomials in their simplest forms, that is, a form written in prime numerical value. The process of factoring a polynomial with four terms is called factor by grouping. With all factoring problems, the first thing you need to find is the greatest common factor, a process that is easy with binomials and trinomials but can be difficult with four terms, which is where grouping comes in handy.
Examine the expression 10x^2 – 2xy – 5xy + y^2. It is read 10 x-squared minus 2xy minus 5xy plus y-squared. Draw a line between the middle two terms, thereby dividing the problem into two groups of terms: 10x^2 – 2xy and 5xy + y^2.
Find the greatest common factor in the first binomial, 10x^2 – 2xy. The GCF is 2x. Two goes into 10, five times, and into 2, once, and x goes into both terms once.
Divide each term in the first group by the GCF, writing the factors inside the parentheses and leaving the GCF out in front of the parenthetical monomial expression: 2x(5x – y).
Bring down the subtraction sign from the beginning expression: 2x(5x – y) -.
This sign is important because if you forget it, you will not know what sign to use in the factoring of the second monomial.
Find the GCF in the second group of terms, 5xy + y^2. In this case, y goes into both. Divide the second term by the GCF and write the monomial in parenthetical form: y(5x – y). The whole expression now should read: 2x(5x – y) – y(5x – y). Notice both parenthetical monomials match. This is important; if they do not match, the factoring process is incorrect.
Rewrite the terms using parenthetical notation. The first monomial is the terms within the parentheses and the second monomial is the two outside terms. The answer to the factoring polynomials with grouping example is (5x – y)(2x – y).
Multiply the monomials with the FOIL method to double-check your work. Multiply the first terms, (5x)(2x) = 10x^2. Multiply the outside terms, (5x)(–y) = -5xy. Multiply the inside terms, (-y)(2x) = -2xy. Multiply the last terms, (-y)(-y) = y^2. (Remember two negatives multiplied together equal a positive).
Rewrite the multiplied terms to see if they match those in the original polynomial: 10x^2 – 5xy – 2xy + y^2. Even though the middle terms are switched because of the FOIL method, they are still the same numbers from the original polynomial.