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

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.

Examples
Rewrite the following algebraic expressions as addition.
a)
b)
SECTION 17.2: MULTIPLYING
MONOMIALS BY MONOMIALS
DEFINITIONS
Monomial: a number, a variable, or a number
times one or more variables.
ex.1) 5 ex. 2) x ex. 3) 5x ex. 4) xyz ex. 5) 11yxzm
Factor: a "math expressions" that is multiplied.
ex. 1) (2)(3)(5) 

2, 3 and 5 are factors 
ex. 2) 5xyz 

5, x, y and z are factors 
ex. 3) 7(x + 2)(x  4) 

7, (x + 2) and (x  4) are factors 
MR. L'S COMMENTS:
As you know by now factors can be reordered and
WILL NOT change the result of multiplication.
ex) (7)(3) = (3)(7)
As an aside, you are suppose to know this same principle
can be applied to terms that are ADDED.
ex) 5 + 3 = 3 + 5 
You know from a previous section what
5s^{4} 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,
s^{4} 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.

RULE FOR MULTIPLYING
EXPONENTIALS
where a, b, m and n are real numbers.

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

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
_________________________________________________
Examples of Multiplying
Monomials:
ex. 1)
comment: There are 3 factors in 35xy. WHY?
ex. 2) (2t^{3})(4t^{3}) = (2)(4)(t^{3})(t^{3}) = 8(ttt)(ttt) = 8t^{6}
comment: There are 7 factors in 8t^{6}. WHY?
ex. 3)
comment: Notice I used the rule for multiplying exponentials in this problem. WHY?
Also, there are 14 factors in 56g^{13}. WHY?
ex. 4)
comment: Notice I used the rule for multiplying exponentials in this problem. WHY?
Also, there are 235 factors in (3/10)h^{234}. WHY?
SECTION 17.3: MULTIPLYING
MONOMIALS BY BINOMIALS AND MORE
Consider the following problem: The operation between h^{3} 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 k^{3} through each term of the binomial .
Let's try another problem, again using the distributive property
ex) Multiply .
ex.)
Step 1) )
Step 2) 8u^{7}(6u^{9} ) + 8u^{7}(2u^{12} ) = 48u^{16} + 16u^{19}
Step 3)
Comment: These terms cannot be added because they are not "like terms."
Therefore:
!!!!! IMPORTANT !!!!!

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.

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."

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.

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
BINOMIAL TIMES A BINOMIAL RULE 
STEP 1) Change all subtraction to addition based on the mathematical fact that subtraction is really addition. 
STEP 2) Multiply the first term of the first binomial by the second binomial using the distributive property. 
STEP 3) Multiply the second term of the first binomial by the second binomial using the distributive property. 
STEP 4) Add the results from step 2 and step 3, then simplfy like terms. 
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) =
14x^{2 }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) =
SECTION 17.5: MULTIPLYING
POLYNOMIALS BY POLYNOMIALS
Would you know how to multiply ?
You are being asked to multiply a polynomial times a polynomial. Here is the
rule:
MULTIPLYING A POLYNOMIAL BY A POLYNOMIAL 
STEP 1) Change all subtraction to addition based on the mathematical fact that subtraction is really addition. 
STEP 2) Multiply each term of the first polynomial by the second polynomial using the distributive property. 
STEP 3) Add the results from step 2 , then simplfy like terms. 
ex. 1) Multiply the two polynomials:
Step 1)
Step 2)
Step 3)
Try this on your own:
