Binomial Expansion

Introduction to the FOIL Method

Lesson Overview

Instructor Presentation

Introduction

We already learned how to use the Distributive Property for multiplication in cases like the following:


3 (x + 5) = 3x + 15

Today we'll talk about cases where we're multiplying two equations (two binomials):


(x + 5) (x + 3)

Goals

Students will be able to use the FOIL method to expand binomials

Students will work through practice problems to get comfortable with this process through repetition

Instructor Presentation

FOIL Method

"FOIL" is an abbreviation for a common way to multiply binomials

F     →  First
O    →  Outer
I      →  Inner
L     →  Last

Instructor Presentation

FOIL Method

Let's work through the steps one by one:

Instructor Presentation

FOIL Method

Instructor Presentation

FOIL Method

Instructor Presentation

FOIL Method

Instructor Presentation

FOIL Method

The last step is to add up each part:

Instructor Presentation

FOIL Method

Let's look at a different example with a negative sign rather than all positives:

Concept Exploration

Use the sliders in the example below to answer the following questions:

Manipulate[
Column[{
Text[Style[
Row[{
"(x ",If[a ≥0, " + ", " - "],ToString[Abs[a]],")"," (x ",If[b ≥0, " + ", " - "],ToString[Abs[b]],")"
}],
 Large]
],
Grid[{
{Text[Style["F", Large]],"irst   ", Text[Style["O", Large]],"uter   ",Text[Style["I", Large]],"inner   ",Text[Style["L", Large]],"ast   "},
{Style["(x^2)", Large], , Style[b*x, Large],,Style[a*x, Large],,Style[a*b, Large]}
}],
Style[Expand[(x+a) (x+b)], Large]
}],
{a, -10, 10, 1}, {b, -10, 10, 1}
]

(1)  Move the sliders so that both "a" and "b" are negative values, for example a  = -4 and b = -7.  Why is the last term +28 rather than -28?

(2)  If you set the sliders such that a = 3 and b = 5; then set the sliders so that a = 5 and b = 3, should the answer be the same?  Why?

(3)  Move the sliders through all possible cases, what other general things to do notice?

Examples

Practice problems.  

Click on "New Problem", type your answer in the box, then click on "Grade" to see if you are correct.  You can type "
(x^2)" by either pressing the "Control" and "6" keys (then right arrow to go back to normal text) or the "Shift" and "6" keys together.

Work on this until you can get 10 correct in a row.

Panel[
DynamicModule[
{eq=(x+3) (x+4),answer="enter answer here", grade="ready to grade"},
Column[{
"Equation:",
InputField[Dynamic[TraditionalForm[eq], (eq=#)&]],
"Write expanded form here:",
InputField[Dynamic[TraditionalForm[answer], (answer=#)&]],
Button["Grade", checker[eq, answer]],
Button["New Problem", {eq=(x+a) (x+b)/.{a->RandomInteger[{-10,10}],b->RandomInteger[{-10,10}]}, answer="enter answer here", grade="ready to grade"}],
InputField[Dynamic[grade]]
}], Initialization:>(checker[eq_,answer_]:=
If[Expand[eq]===answer, grade=Correct, grade=Incorrect])
]
]

Examples Answer Key

Concept Exploration

(1)  Move the sliders so that both "a" and "b" are negative values, for example a  = -4 and b = -7.  Why is the last term +28 rather than -28?

When multiplying two negatives, the answer is positive:  -4 * -7 = 28

(2)  If you set the sliders such that a = 3 and b = 5; then set the sliders so that a = 5 and b = 3, should the answer be the same?  Why?

Yes, this is an application of the Commutative Property

(3)  Move the sliders through all possible cases, what other general things to do notice?

When either "a" or "b" is zero, the third term reduces to zero.  Any two positive "a" and "b" values will have all positive FOIL values.  Setting a = -b will reduce the middle term to zero in all cases.  There are also patterns when "a" is negative and "b" is positive (or the reverse); if a 

> b in this case, the middle term will always be negative (and the reverse is 

true).

Practice Problems

Self-grading.