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(a+b) (c+d) is the same as
A. a + b(c+d)
B. ac + bd
C. (a+b)c + d
D. (a+b)c + (a+b)d
This question matches the Learning Standard Mathematical Structure:
Compare and contrast the real number system and its subsystems with regard to structural characteristics
Demonstrate the logic of algebraic procedures and their interrelationship with geometric ideas and concepts
The correct answer is D.
Where did that come from? I bet you looked at the question and said, "Oh, this is easy. I just use the FOIL method to multiply two binomials."
So you figured the answer was ac+ad+bc+bd.
But, that's not a choice. . . Or is it?
Remember the Commutative and the Associative Properties of Multiplication and Addition? Aha! The Commutative Property lets you multiply or add in any order. The Associative Property lets you change the grouping when you multiply or add.
Remember the Distributive Property?
a(b+c) = ab+ac
Let's take ac+ad+bc+bd and change the order of the terms so the terms having c's are grouped together, and the terms having d's are grouped together.
(ac+bc) + (ad+bd)
Now, let's apply a little Distributive Property.
(c+d)a + (b+a)d
So, you were right! It is a FOIL problem, you just had to apply your knowledge of mathematical structure.
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