ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 17: MULTIPLYING POLYNOMIALS

SECTION 17.1: SUBTRACTION IS A FORM OF ADDITION

In an earlier section you learned that subtraction is a form of addition. For example, 5 -7 is actually 5 + -7 which is equal to -2. If you wish, you can rewrite all subtraction as addition if it enables you to perform certain mathematical processes more efficiently. Many students have difficulties when subtraction is mixed with a signed numbers so making this switch from subtraction may be useful. But in order to master algebra and upper level mathematics you must internalize this process so that you don't have to explicitly makes these changes on your work paper.

For example, 3x - 5 can be rewritten as 3x + -5. This ability will be useful in this section.

THE RULE FOR WRITING SUBTRACTION AS ADDITION
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Change subtraction to addition, then change the sign of the number that

was to the immediate right of the subtraction symbol to its opposite.

In other words, a positive number would become negative and a

negative number would become positive.
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Examples

Rewrite the following algebraic expressions as addition.

a)

b)

SECTION 17.2: MULTIPLYING MONOMIALS BY MONOMIALS

You know from a previous section what 5s⁴  means 5(s)(s)(s)(s) since the power applies to the number or variable immediately to the left of the power. In this case s is the "base." You also know that variables or numbers that are raised to a power are called "exponentials." So, s⁴  would be called an "exponential."

Let's consider what means:

Note that I "expanded" into factors. Because of the properties of multiplication, in this case the commutative property of multiplication,.I am able to move the factors around to suit my purposes.

Consider the following:

From the above examples I can infer the following rule for multiplying exponentials which in this case are monomials.

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RULE FOR MULTIPLYING EXPONENTIALS

where a, b, m and n are real numbers.

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Essentially what we are doing above is multiplying "monomials."

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MORE EXAMPLES OF MONOMIALS

ex. 1) is equal to -6xx which is made up of 3 factors.

ex. 6) is equal to 12xxyyyyy which is made up of 8 factors

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Examples of Multiplying Monomials:

ex. 1)
comment: There are 3 factors in 35xy. WHY?

ex. 2) (-2t³)(-4t³) = (-2)(-4)(t³)(t³) = 8(ttt)(ttt) = 8t⁶ 

comment: There are 7 factors in 8t⁶. WHY?

ex. 3)
comment: Notice I used the rule for multiplying exponentials in this problem. WHY?
Also, there are 14 factors in -56g¹³. WHY?

ex. 4)
comment: Notice I used the rule for multiplying exponentials in this problem. WHY?
Also, there are 235 factors in (-3/10)h²³⁴. WHY?

SECTION 17.3: MULTIPLYING MONOMIALS BY BINOMIALS AND MORE

Consider the following problem: The operation between h³  and is multiplication. Why? What we have here, , is a monomial times a binomial. A "binomial" is two monomials that are added(or subtracted).

Now based on the distributive property I have to multiply k³  through each term of the binomial .

Let's try another problem, again using the distributive property

ex) Multiply .

ex.)

Step 1) )

Step 2) -8u⁷(-6u⁹ ) + -8u⁷(-2u¹² ) = 48u¹⁶ + 16u¹⁹

Step 3)
Comment: These terms cannot be added because they are not "like terms."

Therefore:

!!!!! IMPORTANT !!!!!
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Ideally you are expected to change all subtraction to
addition in your head. Teachers expect that
you can do this. I am changing subtraction to addition
in order to show you the algebraic rule that justifies
the symbol manipulations.
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Now what about the situation when you multiply a monomial by a "trinomial" which is 3 monomials added or when you multiply a monomial by more than 3 monomials. At this point we will define a "polynomial."

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POLYNOMIAL

A polynomial is one or more monomials that are added.
AND
You will use the same rule you use for multiplying monomials by
binomials when multiplying a monomial times a polynomial.

Comment: The above definition of a polynomial is NOT exact and an
over simplification. This will suffice for now.
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ex.) Multiply the monomial by the polynomial:

Comment: Notice how I did not leave a negative number to the right of addition. Remember, addition followed by a negative is subtraction; and subtraction followed by a negative is addition.

SECTION 17.4: MULTIPLYING BINOMIALS BY BINOMIALS

You have been taught in previous classes that the above technique for multiplying a binomial times a binomial is called the F.O.I.L. technique.

ex 1. Multiply

Step 1) Note how I changed all subtraction to addition.

Step 2) produces (2x)(-7x) = -14x² and (2x)(8) = 16x.

Step 3) produces (-3)(-7x) = 21x and (-3)(8) = -24.

Step 4)

ex 2. Multiply

Step 1) Note how I changed all subtraction to addition.

Step 2) produces and

Step 3) produces and

Step 4) =

ex 3. Multiply

Step 1) Note how I changed all subtraction to addition.

Step 2) produces and

Step 3) produces and

Step 4) =

Would you know how to multiply ? You are being asked to multiply a polynomial times a polynomial. Here is the rule:

ex. 1) Multiply the two polynomials:

Step 1)

Step 2)

Step 3)

Try this on your own: