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Multiplication of polynomials and evaluating expressions

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1. How would you teach the multiplication of polynomials?

2. What four steps should be used in evaluating expressions? Could these steps be skipped or rearranged? Explain your answersr.

3. Do you always use the property of distribution when multiplying monomials and polynomials? Explain why or why not. Also what type of situations would distribution become important?

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Solution Preview

1) Multiplication of polynomials is VERY similar to multiplication of whole numbers. Take, for example, a simple one:

111x77

You would write the problem like this

111
x77
-----
777
+7770

So what you have done is broken it up into several addition steps. We can do the exact same with polynomials:

Take (x^2+2x+7)*(x+2)

Arrange it exactly the same!
x^2+2x+7
x+2

Now generate addition steps. We go from right to left, just like in the multiplication of whole numbers. Start by multiplying 2 to (x^2+2x+7)

2*(x^2+2x+7)=2x^2+4x+14

Next, multiply x to the polynomial

x*(x^2+2x+7)=x^3+2x^2+7x

So now our work looks like this:

x^2+2x+7
x+2
--------------
2x^2+4x+14
+ x^3+2x^2+7x
=x^3+2x^2+2x^2+4x+7x+14

We can combine like terms to simplify, ending ...

Solution Summary

This solution answers the three questions posed by the original student, so the solution itself seeks to explain these processes in as much detail as possible. It is approached from the perspective of helping a parent teach their child the math, so it is presented in solution form in as simple a manner as possible, creating a thorough set of rules and hints for solving the multiplication of polynomials and using order of operations. Also, we answer the question, "Why do we use the FOIL method when multiplying binomials?" This solution is a real tour de force!

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