Unit on Factoring
8.4.1 Common Monomial Factors
In this unit students learn to factor expression such as 2x² + 4x for which the greatest common factor is 2x. The factored expression is 2x(x + 2). Also covered are:
1) Factoring by grouping in which ax + by + ay + bx is factored into (ax + bx) + (ay + by), and then (a + b)x + (a + b)y, which factor into (a + b)(x + y).
2) The factoring of two squares such as x² - y² which becomes (x – y)(x + y) which follows the product of the sum and difference of two variables.
3) The factoring of perfect trinomial squares such as x² + 4x + 4, as well as using the short-cut FOIL method for squaring a binomial
4) Factoring trinomials in the form ax² + bx + c, for a =1, and the more general case when a is greater than 1.
5) Complete factorization in which an expression such as 2x² + 10x + 12 is factored as 2(x+3)(x + 2).
6.) Solving quadratic equations by factoring; e.g., x² + 5x + 6 = 0
Because this is a lot of information presented, students can become confused. What was once straightforward such as factoring 2x² + 4x gets left unfactored because students have become used to factoring trinomials. Conversely an equation like x² + 5x + 6 = 0 is factored as x(x + 5) = -6, and is left unsolved. Interleaving of the various types of factoring is therefore essential in this unit, so that students are not left with mastery of the latest technique learned, with the others forgotten.
This particular lesson follows a lesson on what factoring is, and Greatest Common Factors, some of which are included in this lesson’s Warm-Ups. Students have learned about GCF’s sometimes as early as fifth grade. The prior lesson expands this so students must find the GCF for 5b, and 40bc (Answer: 5b).
Most problems with factoring involve an intuitive approach to finding the greatest common factor between two numbers such as 8 and 12 (for which it is 4). Students also know when the greatest common factor is the smaller of two numbers when the second number is a multiple of the first, such as 4, 12, or 6, 18.
Warm-Ups.
Common Monomial Factors. Problem 5 of the Warm-Ups presents a distribution problem. “You’ve been doing distribution problems for a while, starting in seventh grade. In Problem 5 we had a distribution for which the answer is ax + bx. We can also start with ax + bx as the final product of a distribution and work backwards to find out what happened before it was distributed. We already know the answer; x was distributed across a + b. It is the common monomial factor of ax + bx.”
“Now I’ll ask you if you can find the common monomial factor of 2x + 10. This is a final distribution product. What did it look like before the distribution?”
Students usually tell me the answer quickly: 2.
“It is 2. So how would I write that pre-distribution?” I have them write on their mini-whiteboards—most get it right off: 2(x + 5).
“What you’re really doing is the opposite of what you do in distribution. To distribute the 2, you multiply it by the variables and numbers inside the parentheses. When we factor it, what are we doing?”
I will hear “divide”.
“Yes. We are ‘un-multiplying’, otherwise known as dividing; we are dividing 2x and 10 by 2. We are undoing the distribution by dividing by the greatest Common Monomial Factor of the polynomial.”
I have them write this definition in their notebooks:
The greatest common monomial factor of a polynomial is the common monomial factor having the greatest numerical coefficient and the greatest degree.
I have them paste in to their notebooks the steps for factoring a polynomial:
Examples. (These are taken from the first homework problems assigned and worked together as a class)
Homework. The homework problems include binomials and trinomials all with common factors. Problems that require factoring by grouping are discussed in the subsequent lesson).