Notes on Multiplying Polynomials - Page I
Name_________________________
MM1A2. Students will simplify and operate with radical expressions, polynomials,
and rational expressions.
c. Add, subtract, multiply, and divide polynomials.
--------------------------------------------------------------------------------------------------------------------A rectangle has an unknown length and a width of 4 units. If the length is increased by 7 units to
create a larger rectangle, write a simplified algebraic expression for the area of the new, larger
rectangle.
x
7
The figure to the left displays the scenario described
above. In order to find the area of the new, larger
rectangle, one must multiply width times length.
4
4
What is the width of the whole rectangle?
original
x
7
What is the length of the whole rectangle?
Therefore, to find the area, one must simplify 4( x  7) .
This presents a problem for us because the "length" is a polynomial ( x  7 ), and we have not yet
learned how to multiply polynomials. So, let's devise another way to find the area.
To find the area of the whole rectangle, simply find the areas of the smaller rectangles and add them
together. See if you can do that now.
To find the area of the first rectangle, you multiplied 4 times x .
To find the area of the second rectangle, you multiplied 4 times 7.
This same process could be done with the original algebraic expression: 4( x  7) . In actuality,
the number on the outside (4), gets multiplied by both terms on the inside.
4( x  7)
= 4x  28
This property, as previously taught in middle school, is known as the distributive property.
Distributive Property: a(b  c)  ab  ac
-------------------------------------------------------------------------------------------------------------------Simplify: x (2 x  5) . Draw arrows to demonstrate how the distributive property works.
--------------------------------------------------------------------------------------------------------------------Simplify the expressions.
1. 3 y(4 y 6  2 y 3 )
2. 5 z 2 (3z  1)
3. 2 x 4 ( x 2  3x  9)
4. (a  1)  11a
5.
A rectangle has an unknown length and width. If the length is increased by 6 units to
create a new, larger rectangle, write a simplified algebraic expression for the area of
the new rectangle.
--------------------------------------------------------------------------------------------------------------------Suppose a square has an unknown length. If the length is increased by 5 units, and the width is
increased by 7 units to create a new, larger rectangle, write a simplified algebraic expression for the
area of the new rectangle?
x
x
5
The figure to the left displays this scenario. Label the original
square. Once again, one could find the area by multiplying length
times width.
What is the length of the new rectangle?
7
What is the width of the new rectangle?
Thus, to find the area, one must multiply ( x  5)( x  7) . As before, in order to learn how to perform
this multiplication, find the area of the whole rectangle above by finding the areas of the four smaller
rectangles and adding them together.
To find the area of the upper-left rectangle, you multiplied x times x.
To find the area of the bottom-left rectangle, you multiplied x times 7.
To find the area of the upper-right rectangle, you multiplied 5 times x.
To find the area of the bottom-right rectangle, you multiplied 5 times 7.
So, essentially, in the expression ( x  5)( x  7) , you multiplied each term in the first polynomial by
each term in the second polynomial. Draw arrows to show how this process works.
( x  5)( x  7)
This multiplication process from this problem is also known as FOIL (First, Outer, Inner, Last).
--------------------------------------------------------------------------------------------------------------------Simplify ( y  3)( y  10) .
Notes on Multiplying Polynomials - Page II
Name_________________________
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.
--------------------------------------------------------------------------------------------------------------------6. (2d  1)(d  8)
7. (k 2  5)(4k  11)
8. ( y  4)( y 2  7 y  10) .
--------------------------------------------------------------------------------------------------------------------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)
ab2 (6a  44ab2 )
15.
16.
(c  1)(c 2  c  1)
--------------------------------------------------------------------------------------------------------------------For Questions 17-18, find the area of the entire figure.
17.
18.
4x
2
3x
9
--------------------------------------------------------------------------------------------------------------------19.
A rectangle has a length of 8 units and an unknown width. If the length of the rectangle
is increased by an unknown amount, write a simplified algebraic expression for the area
of the new rectangle.
20.
A square has an unknown length. If its length is increased by 3 units, and its width is
increased by 9 units to create a new, larger rectangle, write a simplified algebraic
expression for the area of the new rectangle.
21.
A rectangle has a length that is three times its width. If the length is increased by 5 units
to make a larger rectangle, write a simplified algebraic expression for the area of the
new rectangle.
22.
A rectangle has an unknown length and width. If both dimensions are increased by 12
units to create a larger rectangle, write a simplified algebraic expression for the area of
the new rectangle.
--------------------------------------------------------------------------------------------------------------------1. 12a  8
2. 90b3  9b 2  63b
3. 56c3  16c
4. 7 d 2  7 d
5. 10 x5  2 x 2
6. 30 y 5  65 y 2
7. z 2  6 z  8
8. 3m 2  16m  21
9. 12mn  54m  2n  9
10. r  25
11. 6 p  4 p  3 p  2
12. 7 x 2  30 x  8
13. 4 x3  3 x 2  2 x  15
14. 2 y 3  22 y 2  61y  35
15. 6a 2b 2  44a 2b 4
16. c3  1
17. 2 x 2  16 x  30
18. 12 x 2  42 x  18
19. lw  8w
22. lw 12l 12w 144
20. l 2  12l  27
21. 3w2  5w
2
3
2