The multiplication of monomials and polynomials is an important skill in algebra and subsequent math courses. Students seeing that they are going to learn how to multiply two polynomials—starting with two binomials—believe that it is going to be a difficult procedure. In fact, students find it quite easy and often anticipate the method before I finish my presentation of its genesis.
I do not present the short-cut for multiplying two binomials (referred to as FOIL) in this lesson. The method presented in this lesson will work for any two polynomials. Multiplication of polynomials other than binomials do not lend itself to the FOIL technique, so it is important that students know the general method from which the short-cut for multiplying two binomials is derived.
Students may still be making common mistakes such as x² + x² = x⁴ or x² + x² = 2x⁴, as well as x² + x³ = x⁵. I therefore include problems in Warm-Ups to flush out such errors and disabuse students of these bad habits.
Warm-Ups.
In Problem 1 and 3, students may add exponents. Like terms may be combined by adding or subtracting, but exponents remain as is. Students may need to be told that a can be distributed just the same as if it were on the left hand side, which is how they’re used to seeing it. For Problem 4, possible prompts would be to write on the board: Distance for 2 hours + distance for 4 hours = 1150 “What’s the formula for distance when we know rate and time?” “What is the time for the first leg of the trip? For the second? How do we represent speed for the first leg? For the second?” Problem 5 is an “or” situation; there is no number that satisfies both inequalities.
Multiplying Two Binomials. By virtue of Problem 2 of the Warm-Ups (and some of the problems in their homework), students know that the number or variable to be distributed can be on either side of the parentheses. I use this to motivate the procedure for multiplying two binomials.
“In Problem 2 of the Warm-Ups, we had (x + 2)a. And you solved it just fine. Let’s do it again, with a different binomial.”
“Now distribute it.” If all goes well, and it usually does, they will get this.
“We’re going to learn how to multiply two polynomials, starting with two binomials. Let’s start with “(2x +3)(4x + 5).”
I pause for dramatic effect and they wait as if I’m a magician about to reveal an ancient secret.
“Let’s let z which we just distributed across (2x + 3) equal (4x + 5). And I’m going to substitute (4x + 5) into (2xz + 3z). I’ll do the first one. 2x (4x+5). Now you do 3z.”
Most students get 3(4x+5) which I find with a quick check of notebooks. “Now let’s put it together.”
“Now distribute each one and combine like terms. Do it in your notebooks.”
“What you’ve just done is distributed the binomial (4x + 5) across (2x + 3), just like you distributed the z across (2x + 3). Remember, z is a variable; it represents a number. We can think of (4x + 5) in the same way. It’s a number, which we can distribute. Let’s see this with another example.”
“If we think of (x + 2) like z, we can distribute it to the x and the 7. Let’s think of (x + 2) as z first. Then we’d get this.”
“Let’s plug in (x + 2) for z now.”
Treating an expression with two or more terms as a single entity is called “chunking”, just as we think of an area code of a phone number is a single entity, rather than memorizing each digit of the area code as we would a new phone number. By now, most students are chunking the (x +2) binomial and seeing that it’s treated as a single variable which is then distributed. I have them write the rule:
To multiply one polynomial by another, use the distributive property: multiply each term of one polynomial by each term of the other, and then add the products.
Examples:
Multiplying a Binomial by a Trinomial. The same method applies when multiplying a binomial and a trinomial. I write on the board:
“In this case, what do we ‘chunk’ and how to we distribute it?”
Some students will take the trinomial and distribute it across (x + 5), while others will want to distribute the (x + 5) across the trinomial. If I get both suggestions I’ll have each student write it out at the board to see that the answers are the same. I check carefully as they do it to ensure there are no mistakes along the way so the result is in fact the same.
I will have maybe two more and then start in on the homework.
Homework. Problems will consist mostly of multiplying two binomials, but will also include binomial times a trinomial and a challenge problem of two trinomials. Also included are problems where a monomial is multiplied by a binomial which they learned in the previous lesson.