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In this lesson we are going to
expand our
discussion on adding monomials to include
larger algebraic expressions called polynomials. |
The prefix "poly" means
"many". So a polynomial is an
expression made up of many terms.
Remember that a term can be a variable,
a number (called a constant), or a variable with a
coefficient (a number attached to the front of a variable).
For example, x, 13, 5y are all terms.
When you connect two, or more terms, with either
a "+" or a "-" sign you create different types of
polynomials.
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An expression made up of 2 terms
is called a binomial.
3x + 5, 2y -15,
x + 6f are examples of
binomials. |
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An expression made up of 3 terms is called a
trinomial.
4a+7y-2z,
3x2-7x+4 are examples of
trinomials. |
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When 4 or more terms are connected we
simply call them polynomials. |
Now let's see
how adding and subtracting
polynomials is done.
Let's look at an example for addition:
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Add: 3x+6y+7z
and 5x-4y+9z
Solution:
First...remember that you can only add like terms!
Like terms are terms which contain the same variable(s).
That means that in our example above we will combine only those terms in
the first polynomial which are like other terms in the second
polynomial.
3x+6y+7z
+ 5x-4y+9z
=
8x+2y+16z
Notice that the
variables did not change in any way...we only added or subtracted the
coefficients! |
Another Example....
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Add: 3x+6y+7z
and 5x-4y+9z
Solution:
First...remember that you can only add like terms!
Like terms are terms which contain the same variable(s).
That means that in our example above we will combine only those terms in
the first polynomial which are like other terms in the second
polynomial.
3x+6y+7z
+ 5x-4y+9z
=
8x+2y+16z
Notice that the
variables did not change in any way...we only added or subtracted the
coefficients! |
One more example
of adding....
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Add: -4x3+6x2-8x-10 and 7x3-4x2+9x+3
Solution:
This time let's see what happens when we put the polynomials inside
parentheses.
(-4x3+6x2-8x-10) + (7x3-4x2+9x+3)
To remove the parentheses we must use the
distributive property.
There is nothing in front of the first parentheses so we can just drop
(remove) them.
In front of the second parentheses is a "+" sign.
When we distribute the sign through the parentheses we multiply each of
the signs inside the parentheses by the "+" sign that is outside and the
result is:
(-4x3+6x2-8x-10) + (7x3-4x2+9x+3)
-4x3+6x2-8x-10+7x3-4x2+9x+3=
3x3+2x2+x-7
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Now let's look
at subtraction....
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Subtract: 8a+5b-6c from 10a+8b+7c
Solution:
We must use parentheses for subtraction!
Remember the polynomial after
the word "from" is placed first in the subtraction problem.
(10a+8b+7c) - (8a+5b-6c)
Clear the parentheses by distributing the signs...
10a+8b+7c-8a-5b+6c
Then combine the like terms...
10a+8b+7c-8a-5b+6c
2a+3b+13c |
Let's look at 1 more of these...
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Perform the indicated operation:
(-3x2+4x-11) - (-6x2-8x+10)
Distribute the signs in order to clear the parentheses...
-3x2+4x-11+6x2+8x-10
Combine like terms:
3x2+12x-21 |
For
Practice
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