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3.2.3 Multiplying Binomials and the distributive property
In addition to using Algebra tiles and Punnett Squares to help us multiply expressions, the acronym FOIL is
also helpful..
Example 1: Use a Punnett Square to solve (3x + 5)(2x – 7)
3x
5
6x2
-21 x
10x
-35
6x2 + 10x -21x -35
2x
-7
6x2 -11x -35
Using the Foil method:
a)
b) Solve using Foil: (2x + 1)(X – 3)
Solve using a Punnett Square (2x + 1)(X – 3)
c) Solve using a Punnett Square: (x + 4)(x + 2)
4)(x + 2)
Solve using a Punnett Square: (x +
Distributive Property
We use the Distributive Property to write a product of expressions as a sum of terms. The Distributive
Property states that for any numbers or expressions a, b, and c, a(b + c) = ab + ac.
For example, 2(x + 4) = 2x + 2*4 = 2x + 8. We can demonstrate this with algebra tiles or in a generic rectangle.
Other examples:
3-54. For each of the following rectangles, find the dimensions (length and width) and write the area as the
product of the dimensions and as the sum of the tiles. Remember to combine like terms whenever possible.
a)
b)
3-55. Area as a product is given below. Write an equivalence statement for its area as a as a sum. See
diagram for further explanation. You may use any method you prefer. Show your work for credit.
a. (x + 3)(2x + 1)
b. 2x(x + 5)
c. x(2x + y)
d. (2x + 5)(x + y + 2)
e. (2x + 1)(2x + 1)
f. (2x)(4x)
g. 2(3x + 5)
h. y(2x + y + 3)
3-56. Use the Distributive Property to simplify:
a) 2x(6x + 5)
c) 3y(4x + 3)
b) 6(4x + 1)
d) 7y(10x + 11y)
3-57. CLOSED SETS
Whole numbers (positive integers and zero) are said to be a closed set under addition: if you add two whole
numbers, you always get a whole number. Whole numbers are not a closed set under subtraction: if you
subtract two whole numbers, you do not always get a whole number: 2 – 5 = –3 (–3 is not a whole number).
Investigate with your team whether the integers are a closed set under addition, and whether the integers are
a closed set under subtraction. Give examples.
If you find that integers are closed under either of the operations, can you explain how you know they are
closed for all integers?
Read the Math Notes in the box and answer the questions below.
Are polynomials a closed set under addition?
Are polynomials a closed set under subtraction? That is, if you add or subtract two polynomials, will you
always get a polynomial as your answer?
Give examples and explain how you know your answer is always true.
HW 20 3-58-3-63