= twice + twice. = lost + lost + lost keys

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Alexander Kelley
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1 THE DISTRIBUTIVE LAW DEFINITION: one term By a term we mean any one number, or variable, or a product and/or quotient of several numbers and/or variables. The key is the absence of any addition or subtraction in a term. For example, 3xy is one term while 3x+y is NOT a term since it is not the product or quotient of several numbers, in fact it is the sum of two terms. IDEA Thus far, our axioms have come in pairs. We have met the Times Table as well as the Addition Table, Associativity for Addition as well as Associativity for Multiplication, similarly two commutativity axiom one for addition and one for multiplication. This will be a reoccurring theme, yet the Distributive Law is an exception. We will NOT learn two different distributive laws. The distributive law is, thus far, the only axiom which combines a statement about both binary operations, addition and multiplication. AXIOM: DISTRIBUTIVE LAW < DL > Having met the definition of term, we can begin to state our axioms in terms of terms. That is, unless stated otherwise, these axioms will be adopted not just for the natural numbers, N, but for all other families of numbers we will soon meet. Even more generally, they will be adopted for any type of term. A basic illustration of the distributive law is the following: ) = A more general illustration would be one were we see it applied not just to natural numbers but any sort of term such as stuffblah + potato) = stuff blah + stuff potato where stuff, blah, and potato could be any term or sum of terms. Another example: twice + ) = twice + twice and another, lost + + keys ) = lost + lost + lost keys Moreover, we will aslo accept the distributive law where the sum occurs on the left side such as 2+5)3 = Moreover, we will also accept it when more than two terms are being added, such as ) = Formally, we state it, if A, B, and C represent each a term or sum of terms: and AB + C) = AB + AC < DL > B + C)A = BA + CA < DL > It should be noted that the distributive law does not apply to everything in the world. For example, you will soon learn about functions, such as logs, exponential, or sine and cosine, these functions look like variables but are not. For example And generally, the distributive law does not apply to functions. It is worth emphasizing that we will assume the DL axiom for natural numbers, variables or symbols that represent natural numbers, as well as other families of numbers which we will meet soon. For now, here are some typical uses of the Distributive Law. DL to get rid of Parenthesis We can use the distributive law to get rid of parenthesis. 3x + 1) = 3 x < DL >

2 DL to add Parenthesis We can use the distributive law to get rid of parenthesis. 3 x = 3x + 1) < DL > more generally, or blah x + blah 1 = blahx + 1) < DL > blah + blah = blah + ) < DL > DL to combine like-terms 3x 2 + 5x 2 = 3 + 5)x 2 < DL > = 8x 2 < AT >

3 Some questions to think about 1. How many terms does this expression have? 3x How many terms does this expression have? 3x + x 3. How many terms does this expression have? 3x 1 4. How many terms does this expression have? 3x + xyz 5. How many terms does this expression have? 3x xyz 6. How many terms does this expression have? 3x xyz + 5x 7. How many terms does this expression have? 3x xyz 5x

4 8. The Distributive law would say... 3x + 5x = Solution: 3 + 5)x 9. The Distributive law would say... 9r + 5r = Solution: 9 + 5)r 10. The Distributive law would say... Y 4 + Y 3 = Solution: Y 4 + 3) 11. The Distributive law would say = Solution: 4 + 3) 12. The Distributive law would say... x + 3 = Solution: x + 3)

5 13. The Distributive law would say... x ) = Solution: x The Distributive law would say... x y) = Solution: x y 15. The Distributive law would say... x 2 + 3yz + yr 5) = Solution: x 2 + 3yz + yr The Distributive law would say... x 2 + 3yz + yr 5) = Solution: x 2 + 3yz + yr The Distributive law would say ) = Solution:

6 18. The Distributive law would say... 3x + 5) x 2 + 3yz + yr 5) = Solution: 3x + 5)x 2 + 3x + 5)3yz + 3x + 5)yr The Distributive law would say x) 5 = Solution: NOTHING! the DL say s nothing about EXPONENTS, and...we haven t even learned them yet) 20. The Distributive law would say... sinx + 5) = Solution: NOTHING! the DL say s nothing about functions, and... we haven t even learned them yet) IF sin was a variable which represented a number, we could use DL, but it does not represent a number, it represents a function. 21. The Distributive law would say x) 1 2 = Solution: NOTHING! the DL say s nothing about EXPONENTS, and... we haven t even learned them yet) 22. The Distributive law would say... x 2 + 1)4 + x 2 + 1)3 = Solution: x 2 + 1)4 + 3)

7 23. The Distributive law would say... 53x) = x A. TRUE B. FALSE 24. The Distributive law would say... 53x) = Solution: NOTHING! the DL say s nothing when all we have is ONE term The Distributive law would say... logx + 5) = Solution: NOTHING! the DL say s nothing about functions, and... we haven t even learned them yet).. IF log was a variable which represented a number, we could use DL, but it does not represent a number, it represents a function. 26. The Distributive law would say... A + B)C + D) = Solution: two possible solutions... OR A + B)C + A + B)D AC + D) + BC + D)

sin(y + z) sin y + sin z (y + z)sin23 = y sin 23 + z sin 23 [sin x][sinx] = [sinx] 2 = sin 2 x sin x 2 = sin ( x 2) = sin(x x) ...

sin(y + z) sin y + sin z (y + z)sin23 = y sin 23 + z sin 23 [sin x][sinx] = [sinx] 2 = sin 2 x sin x 2 = sin ( x 2) = sin(x x) ... Sec. 01 notes Some Observations Not Linear Doubling an angles does NOT usually double the ratio. Compare sin10 with sin 20, for example. generally,... sin(2x) 2 sinx sin IS NOT a number If x, y and z are