How to Factor Polynomials by Grouping

Polynomials are algebraic expressions with at least four terms. Students can factor, or break down, these expressions into multiple expressions of three or fewer terms.

Difficulty:

Moderate

Instructions

Put common factors together.

1. Let's look at the example: xy + 3y - 2x -6

2. Re-arrange the terms in the expression so that two consecutive terms have a common factor:xy + 3y - 2x - 6 = xy - 2x + 3y - 6Note that the order of (-2x) and (3y) is switched.

3. Now find the common factor of each of the two consecutive terms:xy - 2x + 3y - 6 = x(y-2) + 3(y-2)

4. Now group the common factors:xy - 2x + 3y - 6 = x(y-2) + 3(y-2) = (x + 3)(y - 2)

Expressions with exponents

1. Here's an example of how to factor a polynomial expression with exponents:x^3 - xy^2 - x^2y + y^3

2. Re-arrange the terms in the expression so that two consecutive terms have a common factor:x^3 - xy^2 - x^2y + y^3 = x^3 - x^2y - xy^2 + y^3Note that the order of (- x^2y) and (- xy^2) is switched.

3. Now find the common factor of each of the two consecutive terms:x^3 - x^2y - xy^2 + y^3 = x^2(x - y) - y^2(x - y)

4. Now group the common factors:x^2(x - y) - y^2(x - y) = (x^2 - y^2)(x - y)

5. Not done yet! Now we need to factor the difference of two squares:(x^2 - y^2)(x - y) = (x + y)(x - y)(x - y) = (x + y)[(x - y)^2]

Tips & Warnings

• For expressions of higher exponents, remember the factors for:

• Difference of Squares: (x^2 - y^2) = (x + y)(x - y)

• Sum of Cubes: (x^3 + y^3) = (x + y)(x^2 - xy + y^2)

• Difference of Cubes: (x^3 - y^3) = (x - y)(x^2 + xy - y^2)

Resources

• Factoring Polynomials YouTube Video
• Factoring Polynomials YouTube Video