Lesson 2.2 Multiplying Polynomials
All the Assignments are listed in SECTION 1. Look in the Navigator box for Assignment Listing.
If you have NOT done the examples below by yourself then do not complete this assignment.
If you cannot do the examples then tell me which ones and ask questions about it.
PLAY the games. If you need a game, ask I might have a new one for this topic. Give me examples! when you ask.
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There are times that exponents do not show up in the text.
If something does not make sense to you then please ask.
Copy and paste the expression into an email. Thanks. SJ
This is a LONG lesson.
WARNING:
Some of the examples are pictures. If you do not see examples 9, 10 and 14 then you need to let me know.
I use diagrams and pictures throughout the course. Sometimes they hide because of my settings, but most often it is your settings or firewall. Let me know if you see a red X or do not see examples 9, 10 and 14.

You must write these examples into your notes for I know you will not remember them all.
We will use these skills in this section and the next one too!
Multiplying a constant and a monomial:
2 examples :
Example 1: 4(2xy^{3}z ) = 8xy^{3}z
Example 2: 7(xyz)= 7xyz
Multiplying a constant and a binomial:
3 examples (notice the add or subtract in the binomial, of course)
Example 3: 3(2xy + 5w) = 6xy + 15w
Example 4: 4(2xyz  3zv) = 8xyz +12zv
Example 5: 7(xyz + 3)= 7xyz+21 Multiplying with a constant was easy.
ARE YOU LOOKING AT EACH of these CAREFULLY? Be a detective. Algebra is about seeing details. Do you know how the problem was worked? LOOK carefully at every letter, number and exponent. You will get done much quicker if you can look for details.
Please carefully watch and write the following:
Multiplying a monomial and a monomial:
2 examples ( notice there is NO add or subtract in monomials)
Example 6 : 5xy(4xy^{2}) = 20xxyy^{2} = 20x^{2}y^{3}
Example 7: 3xy (52x) = 156xxy = 156x^{2}y
Example for monomial raised to a power:
Example 8: (3c^{3}d^{4})^{3}
means to write the (3c^{3}d^{4}) down 3 times and multiply;
(3c^{3}d^{4})(3c^{3}d^{4})(3c^{3}d^{4}) then follow Example 7 or 6.
Let us rearrange these (3)(3)(3)c^{3}c^{3}c^{3} d^{4}d^{4}d^{4}.
This becomes 27c^{9}d^{12}.
Very important thing to see in this lesson:
We do NOT have to have LIKE terms to multiply.
What were we doing when we had to have LIKE terms?
It was adding and subtracting in Lesson 2.1.
Do you know: Add: x + x = 2x and
Multiply: x(x) is x^{2}
Add: x + x^{2} , can not be added so it remains unchanged,
Multiply: x( x^{2}) is x^{3}. You do know that x is really x^{1}.
You might be surprised how many students miss these. (mixingup adding and multiplying rules)

We will return to adding and subtracting LIKE terms.
Please write these examples so that you will recall multiplying.
Try this web site for a little practice.
This is at the web site called QUIA. You do NOT need to join QUIA;
I am the member. You can play this for free.
Multiplying monomials with add/subtract mtr  free http://www.quia.com/quiz/5102741.html

Do Assignment 2.2A before continuing.
Do not go forward until you have done the assignment.
Multiplying a monomial and a binomial:
Since there are LIKE terms inside the above parenthesis
we could add these first and then multiply by the monomial.
The answer would be the same.
Try it. 2xt(9xv + 6xv) = 2xt(15xv) = 30x^{2}tv .
Multiplying a binomial and a binomial requires 2 STEPS.
Watch to see what do in each step. I hope the red letters help..


Example 12: (x + 3)(y + 5) = xy + 5x + 3y +15
Do you know this?
STEP 1:
Multiply the x (first term of first parenthesis) by (y+5).
See x(y+5) = xy + 5x
STEP 2:
Then multiply the +3 (second term of the first parenthesis) by the (y+5) .
See +3(y+5) = +3y +3*5 or +3y +15
That is,
(x+ 3)(y + 5) = xy + 5x +3y +15
Example 13:
WATCH the SIGNS; the sign of a term is on its left side.
(x  4)(w+5) =
Multiply the x by the (w+5) and then multiply the 4 by the (w+5).
x(w+5)  4(w+5) =
Why will the last term be negative?
xw + 5x  4w 20. Do you know that 4 times +5 = 20??
There are 4 terms; none are alike so we will not be able to add or subtract.
Study the above examples carefully. Do you see the pattern?
Write them then try this next one.
Example 14:
x^{2} 4x21 is the preferred answer for (x+3)(x7).
Be sure you understand this.
Again some of this is prior knowledge from Algebra 1.
If you need more email me.
Be sure to include the example that you need more of.
Recall that xxx = x^{3} and x^{2}x = x^{3} also.

What is x^{2} x^{4} ? it is xx xxxx or x^{6}.

Do not confuse it with (x^{2})^{4}

What does this mean? (x^{2})^{4} ?^{ } It means x^{2} x^{2} x^{2} x^{2} = x^{8}

You should take the time to write some of these on a card for quick reference.
Most students will confuse them in a few days.

And also include x^{2} + x^{4} is NOT "addable". They are NOT like terms. Leave it as it is.

This is adding from Lesson 2.1. We do not have LIKE terms so no adding.

But that x^{3} + x^{3} is addable and = 2x^{3.} (We added LIKE terms). DO NOT TOUCH the exponent when adding. You do know where I got the 2?
x^{3} + x^{3 }is really 1x^{3} + 1x^{3} =2x^{3}

How can we get x^{9 } ?

It is NOT x^{3} + x^{3 }NOR x^{3}(x^{3}).

I hope you know that x^{9} is (x^{3})^{3} Write the (x^{3}) down 3 times then multiply to see.
(x^{3}) (x^{3}) (x^{3})= x^{9}
When multiplying we actually add the exponents. That is because the exponent tells us how many of the bases we are to use.
xxx is x^{3}.
xx is x^{2 }or
x^{3} x^{4} = x^{7.}
So what is (x^{3})^{4}?^{ }It is not x^{7} as it is (x^{3})(x^{3})(x^{3})(x^{3}) which is (x^{6})(x^{6}) = x^{12} .

What is x^{9 } It is also x^{4}x^{5} 
or x^{3}x^{6} 
or x^{2}x^{7} 
or (x^{3}) cubed

In an email send me 5 ways to write x^{6. }

Be sure to use only these numbers for the exponents, any way you prefer, to write the answers: 1, 2, 3, 5.

Note: I did not give you the 4 to use, although it can be. 

Try these: (x + 4)(y  5) and (y  9)(y  7)
Look for $$ below for the answers.
Example 15: (x  8)(x  8)= x^{2} 8x  8x +64 = x^{2} 16x+64.
Do not loose the 16x. Why would I say that???
Example 15 1/2: (x8)^{2} is the same as the problem above.
You would be surprised how many tell me the wrong answer.
(x 8)^{2}= x^{2} 16x +64. How did we get that 16x ???
Look at EX. 15.
Want to try out your new skill ???
This is at the web site called QUIA.
You do NOT need to join QUIA; I am the member. You can play this for free.
Go to: http://www.quia.com/quiz/5102726.html
Play more than once and email me your best score ..Hope you do well.
Be sure to return here to the pink box.
Now let's try a binomial times a trinomial.
There will be 6 terms.

Example 16:
(x+7) (x^{2}+6x4) = Multiply the x by the trinomial then multiply the +7 times the trinomial.
x(x^{2}+6x4) +7(x^{2}+6x4) =
xx^{2}+x6x 4x+7x^{2}+7(6x) +7(4)
(See the 6 terms.)
But now let's simplify each term and write it in descending order .
We have LIKE terms
the x6x is really 6x^{2}
and that can be combined with the 7x^{2: }
6x^{2}+7x^{2}=13x^{2}.^{ }
Also the 4x and the +42x=+38x.
= x^{3}+6x^{2}+7x^{2}4x+ 42x28
Combine LIKE terms.
= x^{3}+13x^{2}+38x28.
Now you try one:
Example 17: (y  9)(x^{2} 4y +2) = Multiply the y by the trinomial then multiply the 9 by the trinomial.
yx^{2}y4y+2y9x^{2}+36y18 You should have 6 terms.
= x^{2}y4y^{2}9x^{2}+2y+36y18 COMBINE LIKE terms.
= x^{2}y4y^{2}9x^{2}+38y18
The order of your 6 terms can be different,
but the terms and their sign must be exactly the same.
Remember :
1. the sign of a number is on its LEFT , keep them together.
2.  4xy is the same as  4yx.
3. If there are LIKE terms be sure to combine them.
$$ xy  5x+ 4y 20 or xy 20 5x +4y
$$ y^{2}16y+63
Do Assignment 2.2B only if you have completed the above examples in your notebook.
Be sure you have completed ALL assignments for 2.2.
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