Glossary for Lesson 11: Factoring

Linear Factor

A linear factor is a factor of degree 1.  In other words, it can be written in the form ax+b , where a and b are constants and a is not equal to 0 .  For the quadratic polynomial 2x^2-5x-18 , the linear factors are 2x-9 and x+2 .

Greatest Common Factor

The greatest common factor, or GCF, of two or more numbers is the largest positive integer that divides evenly into all of the numbers (i.e. leaves a remainder of 0).  For example, the GCF of 60 and 18 is 6.
The GCF of two or more polynomials is the largest polynomial of the highest degree that can be factored out of all of the polynomials in question.  For example, the GCF of 3x^2-11x-20 and 6x^2+11x+4 is 3x+4 .

Factoring the Difference of Two Squares

When we have a difference of two squares, like x^2-4 or 9x^2-25 , the expression can be factored according to a certain pattern.  In general, a^2-b^2=(a-b)(a+b) .  For the two examples already mentioned, then, we could factor to have x^2-4=(x-2)(x+2) and 9x^2-25=(3x-5)(3x+5) .

Factoring by Grouping

Factoring by grouping is a technique that can be used to factor certain polynomials that can be written with more than 3 terms.  Let’s look at a couple of examples: x^3+2x^2-5x-10=(x^3+2x^2)-(5x+10)=x^2(x+2)-5(x+2)=(x+2)(x^2-5) If we start out with a polynomial that only has three terms, we can often expand the middle term and then factor by grouping. x^2+3x-40=x^2-5x+8x-40=(x^2-5x)+(8x-40)=x(x-5)+8(x-5)=(x-5)(x+8)

Factoring Sums and Differences of Cubes

Polynomials that are simply sums or differences of cubes factor according to particular patterns.
For any expression of the form a^3+b^3 , we can say that a^3+b^3=(a+b)(a^2-ab+b^2) For any expression of the form a^3-b^3 , we can say that a^3-b^3=(a-b)(a^2+ab+b^2)