Download Notes on Multiplying Polynomials (1)

Notes on Multiplying Polynomials (1)
Name_________________________
Suppose you had to evaluate the following expression: 3(9  5) .
If you follow order of operations, the expression is evaluated 3(9  5)  3(4)  12 .
However, as you may or may not recall, one could also use the distributive property.
Distributive Property: a(b  c)  ab  ac
Draw arrows on the expression at the top in order to show how the distributive property works.
Using the distributive property, the expression is evaluated 3(9  5)  3  9  3  5  27  15  12 .
As you can see, both methods produce the same answer.
--------------------------------------------------------------------------------------------------------------------Hopefully, by this point in the study of polynomials, you have come to the realization that the
same rules and properties that apply to numbers also apply to polynomials. Thus, the distributive
property applies to polynomials as well.
--------------------------------------------------------------------------------------------------------------------Simplify: 3x(9 x 2  4 x) .
Once again, draw arrows to demonstrate how the distributive property works.
So, one multiplies 9x 2 by 3x and 4x by 3x . Here is the result:
3x(9 x 2  4 x)  3x  9 x 2  3x  4 x  27 x3  12 x 2
Therefore, the answer is 27 x3  12 x 2 .
It should be pretty obvious at this point that if a person does not know how to multiply
monomials, then he/she will have no shot at correctly multiplying polynomials.
--------------------------------------------------------------------------------------------------------------------Simplify the expressions.
1. 5 y 2 (4 y 6  2 y 3 )
2. (6b2c  10a 2 )  4ab
3. 2 x 2 (12 x 4  3x3 y  2)
--------------------------------------------------------------------------------------------------------------------It has helped students in the past when multiplying two polynomials to make two columns every term in the first polynomial goes in the first column, and every term in the second
polynomial goes in the second column. Then, multiply every possible combination in column #1
with column #2.
Consider 20 y 4 (3 y 2  4 y  8) .
3y 2
In the space to the right, two columns have been formed.
Every term in the first polynomial ( 20 y 4 ) is in the first
20 y 4
4 y
8
column, and every term in the second polynomial (3 y  4 y  8)
is in the second column.
2
***Notice that the signs accompany the terms when using the column format.***
--------------------------------------------------------------------------------------------------------------------Now, consider (2 y  3)(4 y  5) .
Again, we are only multiplying two polynomials, so set up
two columns in the space to the right as before.
This is really just a more elaborate extension of the distributive
property as discussed before.
In middle school, you may have learned to use FOIL (First, Outer, Inner, Last) when
multiplying two polynomials with two terms each. Feel free to use that here.
--------------------------------------------------------------------------------------------------------------------Use the two-column format or FOIL to simplify: ( x  5)( x  7) .
--------------------------------------------------------------------------------------------------------------------When multiplying two polynomials, multiply every monomial (term) in the first polynomial by
every monomial (term) in the second polynomial, and add the products together.
--------------------------------------------------------------------------------------------------------------------Use the two-column format or FOIL to simplify: ( y  4)( y 2  7 y  10) .
--------------------------------------------------------------------------------------------------------------------It is absolutely imperative for the student to understand...
1)There are many calculations to be done per problem.
2)That translates into more opportunities for mistakes.
--------------------------------------------------------------------------------------------------------------------Simplify: (7b2  4)(3b11  1) .
Notes on Multiplying Polynomials (2)
Name_________________________
One could actually learn to simplify the expression ( x  5)( x  7) by using geometric figures.
Do you recall the formula for finding the area of a rectangle? _________________________
Suppose you had a square with an unknown side length (x).
Now, suppose the length of the square is extended by 5 units, and the width is extended by 7
units in order to form a rectangle. The figure below shows the giant rectangle that is formed.
x
5
Write an unsimplified expression for the area of the entire figure.
x
Find the area of each of the small rectangles.
Add up the areas of each of the small rectangles. Do you notice a
similarity between this answer and the answer on the back of page 1?
7
--------------------------------------------------------------------------------------------------------------------Draw a geometric representation to the right of the following scenario:
A rectangle has an unknown length.
Its width is twice its length.
Then, the length is increased by 4 units.
The width is increased by 3 units
Write a simplified expression for the area of the new rectangle.
--------------------------------------------------------------------------------------------------------------------Homework on Multiplying Polynomials
Simplify.
1.
4(3a  2)
2.
9b(10b2  b  7)
3.
8(7c3  2c)
4.
7 d ( d  1)
5.
2 x 2 (5 x3  1)
6.
(6 y 4  13 y)(5 y)
7.
( z  2)( z  4)
8.
(3m  7)(m  3)
9.
(6m  1)(2n  9)
10.
(r  5)(r  5)
11.
(2 p 2  1)(3 p  2)
12.
(7 x  2)( x  4)
13.
( x 2  2 x  3)(4 x  5)
14.
(2 y 2  8 y  5)( y  7)
15.
ab2 (6a  44ab2 )
16.
(c  1)(c 2  c  1)
17.
Create a geometric representation in the space to
the right for ( x  6)( x  11) .
Inside of each rectangle, write its area.
Finally, write a simplified expression for its area.
18.
Create a geometric representation in the space to
the right for the following scenario:
A rectangle has an unknown length.
Its width is five times its length.
Both the length and the width are extended by 2 units.
Finally, write a simplified expression for its area.
For Questions 19-20, find the area of the entire figure.
19.
20.
4x
3x
9
2