• If you are citizen of an European Union member nation, you may not use this service unless you are at least 16 years old.

View
 

2:1 Polynomials

Page history last edited by Math in a Box - Susan Johnsey gm 1 year, 6 months ago

Lesson 2.1 POLYNOMIALS

 

These lessons are for students that have already studied Algebra for several weeks.  This is not the beginning!

You will need Algebra 1 first semester before beginning this class.  But anyone can try it.   You will learn a lot if you know this one Lesson  very well!

 

STUDY hard and play all the games.   Then complete the ASSIGNMENTS but only if you know it well.

 

If you would like to see this in a wider screen click the "little black triangle" along the right side (below Search Box). 

But do not forget to click it again so that you can see the NAVIGATOR box.

 

Also remember that the WIKI tab above (left)  is the home page for this classroom.  

All lessons and assignments are listed there in the order that they are to be completed.


What is a polynomial? How do we add, subtract, multiply or divide polynomials? 

Please take notes in your notebook and learn the IMPORTANT words.

First we must understand these words :

monomials, terms,  like terms, polynomial, exponents, and variable.

 

The words, exponents and variable, should already be familiar to you.

The Y is the variable and the 2 is the exponent.  

Y2  The Y is called the BASE of the exponent 2.  Ymeans Y multiplied times Y. 

          Yis read "Y with exponent of 2" or usually we say "y squared". 

          We say "squared" only when the exponent is 2.

 

 

  What are Monomials?

 

A monomial is a number, a variable, or a product (multiplication) of numbers and variables.

 

 5 examples of monomials:
  4 or   -3 or   -8xy or    9y2w or      -xy.
   
 When we ADD or SUBTRACT  two or more monomials

  we have a polynomial.

Each of these monomials are then referred to as TERMS of the polynomial.

BE SURE THAT YOU CAN COUNT THE TERMS!

 

2 examples of a polynomial with 3 terms:

x2+3x - 4 or   y-2xy +4t.  These are also called trinomials.

 

2 examples of a polynomial with 2 terms:

 

-x2y +6 or  mt -4s .  These are also called binomials.

 

Be sure you know the above examples for the test.

 

If two polynomials are added then we can

simplify the sum only if the polynomials have at least two like terms.

 

What are  Like Terms ?

 

Examples of like terms (look at these carefully, do you see how they are alike and how they differ):

 

       xz and 3xz and -zx are 3 like terms;

                2x and -1x/2 (half of x) and x42 and -97x are like terms, look carefully at these.

 

Find the terms that are like 5xv:    3vx -7x +5vv -6xv. 

Write them on your paper. Yes, write them. We will check them in  a moment.

 

Find the terms that are like x2y:     8x2y +yx +xy -x2y . 

 

Write them on your paper.  The 2 is an exponent. Be sure you can see that.  

If not let me know or check the web page again if you copied, pasted or printed this. 

                       Some times the exponents FALL when copying and pasting.

 

Find the terms that are like 4zt:  -5zt/3 -4zt +4xt +0tz +1tz +4z2t . 

Write them on your paper.

 

Tricky ones:  5x,   6x2,   7/x,   -5x/2,  -4x/y.    The / means fraction bar or divide.

There are only two that are alike.  Write them on your paper.

 

These are not like terms  4x and 5/x.  

See the x in denominator in the second expression?

that is NOT the same as having x in numerator.  AND the second expression is not called  a monomial because the x is in denominator.  I forgot to mention that earlier so please write that in your notes.  The variable in monomials or any of the -nomials cannot be in denominator.  But the numbers can!

 

You will find the answers below. Look for the $$ then return here.

Be sure that you understand like terms.

Do NOT continue if you do not understand.  

And give an example or something you do not understand.

 

Answers for problems above:

$$  like 5xv:     3vx     -6xv         $$$$$$$$$

$$ like  x2y  :     8x2y     -x2y      $$$$$$$$$

$$  like 4zt :     5zt/3    -4zt   0tz    1tz   $$$$$$$$$

$$ 5x and -5x/2   are LIKE terms    $$$$$$$$$ 

End of answers.

 

 

To add or subtract 2 or more polynomials simply add or subtract the LIKE terms.

You really should already have studied this in a prior algebra course, but here are examples. 

Be sure you can see the colors below.  Let me know if you can not. 

There are three colors.

Example 1 3x+2x2-4+(-4x+5x2)=  

 

GROUP the LIKE terms, but be sure to keep the sign with the term. 

The term's sign stays on its left side.  So this becomes:

3x-4x  +2x2+5x2  -4   now add or subtract like terms.

=-x+7x2-4.

 

Negative sign in front of the parenthesis 

Example 2:    5x3y -4xy -6x   -(3x3y - 4x + 5xy) =    The negative sign in front of the parenthesis tells us to change all the signs of that phrase, 3x3y - 4x + 5xy, and then you remove the parenthesis and remove that negative. 

Change all three signs : -( 3x3y - 4x + 5xy) becomes -3x3y + 4x - 5xy

        5x3y-4xy-6x -3x3y + 4x - 5xy,  I changed the signs of three terms that are inside the (  ) to -3x3y + 4x - 5xy.  

DO YOU SEE?   BE sure you know that the negative in front of the parenthesis means find the OPPOSITE of all the terms inside of the parenthesis.

  THEN remove the parenthesis and that negative.               

  NOW look for LIKE TERMS.       5x3y -4xy -6x -3x3y + 4x - 5xy                                                                              

 Rearrange or look for the LIKE terms. Some students just like to look for the LIKE terms and mark them out as they use them.  Others like to rewrite the problem so the like terms are together.

Do NOT loose their sign; it stays with term:   5x3  -3x3 -4xy -5xy  -6x +4x 

                        = 2x3y -9xy -2x.   

 

Again be sure you understood the sign changes because of the negative in front of the parenthesis.

    I will ask you several questions like this.

 

 

A negative sign in front of a parenthesis will change

the signs of every TERM inside the parenthesis. 

Then the parentheses and the negative in front can be removed. 

 

For Example:

-(3x3y -4x+5xy)  becomes  -3x3y+4x-5xy

 

A positive sign in front of a parenthesis (or NO sign) will NOT change the signs of the  TERMs inside the parenthesis.

 

 

You need to practice the order of Operations

before moving to the next examples.

Do you know the Order of operations? 

                If x = -3 and y= 7 then evaluate x2y -7y +2(x+y). 

Do you know that x2y means xxy?  Can you do the multiplying?  will the answer be positive or negative?

  1.  Many think it is negative; x2y=(-3)2(7) is positive!   63.
  2. Now find the value of the other three terms.   -7y = -7(7) = -49.  

Did you have the negative sign?

  1. And 2(x+y)= 2x+2y = 2(-3)+ 2(7)    did you write this down correctly?

   That is important.  -6+14 =8

If x = -3 and y= 7 then evaluate x2y -7y +2(x+y). 

We will have 63-49 -6+14 = 14-6+14 = 22.

 

  Let me know if you need to review.  What is -x3 if x= -3? or think -xxx if x=-3.  Is it -9 or 9 or -27 or 27?   Tell me why it is +27. Explain well please.______________________________________________________________________.

When you send me questions or email then be sure to copy and paste the problem into the email too.

Play this game.   MUST KNOW WELL the Order of Operations.       

Be sure to record your score.

 



Study this example 3;  Know it well. 

Write it more than once, yes, write it down and watch all the signs.   IT IS A real CHALLENGE!

To try and work this problem with any other steps is wrong.   You must start with -(4x-6).

 

Example 3

3[2x+7-(4x- 6)]+4-7x  =   YOU MUST start inside the [ and ]

 

                         Note the -(4x-6) means opposite of (4x-6) which is -4x +6

3[2x+7 -4x+6]+4 -7x=    We no longer need the  pink negative and the ( and ). 

                       Now LOOK for LIKE terms:  2x+7 -4x+6 becomes -2x +13 .

 

3[-2x +13]+4-7x = Note the 3 is positive and the signs will not change when we multiply with the 3.

-6x+39+4-7x.   LOOK for LIKE terms again.  

This simplifies to  -13x+43. 

 

Now let x= -7.   Evaluate the original problem above and tell me what you get.

3[2x+7-(4x- 6)]+4-7x =_________________________________show some work please and copy and paste the problem, 3[2x+7-(4x- 6)]+4-7x, into the email too. What number did you get?

Now do again but use simplified expression: -13x+43 and x=-7.  Did you get the same number?   You should.

 

 Try this one, it is hard!

3wr+4w+2x-[-x+8w]-(3x-4w) = ?

 

Example 4:

3wr+4w +2x -[-x+8w] -( 3x-4w) =

 

NOTE -[-x+8w] is opposite of [-x+8w] =+x-8w

NOTE -(3x-4w) is opposite of (3x-4w)=-3x+4w .

 

    So we have:  3wr+4w+2x+x-8w-3x+4w =     NOW LOOK for LIKE terms.

                                      

                       3wr+4w-8w+4w+2x+x-3x             = 3wr

 

Do you know why the x-terms disappeared?  and also the w-terms?           

 

 

Watch the exponents.

Sometimes we use the ^ for exponents.  

Look at your keyboard.  It is on your "6 key". 

We write ^3 to mean exponent of 3.

 

x^3     is x with exponent of 3.

 

Below, Simplify each term

then combine like terms.

 

Example 5. (2x)^3 - 3x^3  -7x      

        PLEASE  simplify this and then check it BELOW,  Be sure to know this well for test.

 

             RECALL   2x^3 and (2x)^3 are not the same.  Which one is 8x^3?

 

Example 6.       4[x2 -4(x-3)+(2x)2

 

                                 NOTE the -4(x-3) means to multiply by -4 and get -4x+12 .  

That gives you:

                        4[x2-4x+ 12+ (2x)2]           

           PLEASE finish this one now.

 

Example 7.    x -5(x-2)+2x(4-7)2

 

Note:  

^2 means exponent of 2

^3  means exponent of 3

SEE answers BELOW 

 

Answers are:

 

Example 5

  (2x)^3 - 3x^3  -7x

     =8x^3  -3x^3   -7x

            =5x^3  -7x.

                          Do you know how I got the 8x^3  ??  

Look at the parenthesis and exponent.   That means write the 2x down 3 times and multiply: 2x(2x)(2x).

And do you see why the 3x^3 stayed a 3x^3?

 

Example 6.

4[x2 -4(x-3)+(2x)2] =

4[x2 -4x+12+4x2] =

4[5x2-4x+12] =         

20x2-16x+48

 

Example 7.  

1x -5(x-2) +2x(4-7)2  =            

   I hope you know what MUST be done before the 2x is used. 

         LOOK inside the LAST parentheses!

                              1x -5x+10 +18x =  

Be sure you get the 18x and the +10.

14x+10

 

Do Assignment 2.1  only if you can do these above.  

All the Assignment are listed in SECTION 1.   Look in the Navigator box for Assignment Listing.

If you have NOT done the examples above by yourself then do not complete this assignment.  

If you cannot do the examples then tell me which ones and ask questions about it.   I will be glad to help.                 

 

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

 

Here is a game that will give you review for this lesson as well as the Order of Operations.

 

Working With Algebraic Expressionshttp://www.quia.com/cb/430684.html

 

Comments (0)

You don't have permission to comment on this page.