3.2.4 Using Generic Rectangles to multiply
You have been using algebra tilesand the concept of area, Punnett Squares and the foil method to multiply polynomial expressions. Today you will practice
3-64. Use the Distributive Property to find each product below.
- 6(−3x + 2)
- x2(4x − 2y)
- 5t(10 − 3t)
- −4w(8 − 6k2 + y)
3-65.Write the area as a product and as a sum for the rectangle shown at right.
3-66. Now examine the following diagram. How is it similar to the set of tiles in problem 3-65? How is it different?
3-67.Diagrams like the one in problem 3-66 are referred to as generic rectangles. Generic rectangles (Punnette Squares) allow you to use an area model to multiply expressions without using the algebra tiles.
Find the area of each part and write the area of the whole rectangle as a product and as a sum.
3-68.Multiply and simplify the following expressions usingboth a Punnette Square (generic rectangle) and using the FOIL method.
a)(x + 5)(3x + 2)
b)(2y − 5)(5y + 7)
c)3x(6x2− 11y)
d)(5w − 2p)(3w + p − 4)
3-69. THE GENERIC RECTANGLE CHALLENGE
Fill in the missing dimensions and areas. Then write the entire area as a product and as a sum.
a. c.
b. d,
Review these Properties of Real Numbers
Be prepared for a quiz!
The legal tiles moves have formal mathematical names, called the properties of real numbers.
The Commutative Property states that when adding or multiplying two or more numbers or terms, order is not important. That is:
a + b = b + a For example, 2 + 7 = 7 + 2
a · b = b · aFor example, 3 ·5 = 5 ·3
However, subtraction and division are not commutative, as shown below.
7 − 2 ≠ 2 − 7 since 5 ≠ −5
50 ÷ 10 ≠ 10 ÷ 50 since 5 ≠ 0.2
The Associative Property states that when adding or multiplying three or more numbers or terms together, grouping is not important. That is:
(a + b) + c = a + (b + c) For example, (5 + 2) + 6 = 5 + (2 + 6)
(a· b)· c = a·(b· c) For example, (5 · 2)· 6 = 5· (2· 6)
However, subtraction and division are not associative, as shown below.
(5− 2)− 3≠ 5 −(2 −3) since 0≠ 6 (20 ÷4) ÷ 2 ≠20 ÷(4 ÷2) since 2.5 ≠10
The Identity Property of Addition states that adding zero to any expression gives the same expression. That is:
a + 0 = aFor example, 6 + 0 = 6
The Identity Property of Multiplication states that multiplying any expression by one gives the same expression. That is:
1 ·a = aFor example,1 ·6 = 6
The Additive Inverse Property states that for every numbera there is a number –a such that a+(−a) = 0.A common name used for the additive inverse is the opposite. That is, –a is the opposite ofa.For example, 3 + (−3) = 0 and−5 + 5 = 0
The Multiplicative Inverse Property states that for every nonzero number athere is a number such that A common name used for the multiplicative inverse is the reciprocal. That is,is the reciprocal ofa. For example, .
HW 21 3-70 – 3-75