background of distributive property?
454542
In order to provide you with the theory behind the distributive property, here is a more technical explanation than others are likely to give, but I think you'll be able to follow it:\n\nFirst I need to tell you what an operator is. Suppose we have something like:\n\n5+3=8\n\nThen, "+" would be the operator. Specifically, it's called the addition operator. But if instead we had:\n\n2-1=1\n\nThen "-" would be an operator, and, again, more specifically, it's the subtraction operator.\n\nIn fact, any time we have something which acts between two numbers and produces another number, it's an operator. So, things like multiplication and division are also operators. In fact, we can even make up operators like this:\n\n2o3=10\n\nThere, I made up an "o" operator which happens to produce 10 when you give it 2 and 3. Maybe we would even define the "o" operator like this:\n\naob=2(a+b)\n\nOr, in words, "a operator b equals 2 times the quantity a plus b." So that for 2 and 3 it gives:\n\n2o3=2(2+3)=2x5=10\n\nOk, so now that we know what an operator is, we'll next say the following: operators have properties!\n\nFor instance, the addition operator has the commutative property; i.e. it doesn't matter what order you put the operation in. So, 1+1=1+1. The division operator does not have this property; i.e. 2/1 does not equal 1/2.\n\nThe distributive property is a property that some operators have, while others do not. Actually, the distributive property only comes into play when thinking about combinations of operators. The most common example of two operators that when put together have a distributive property is multiplication and addition. But, in general, an operator is said to have the distributive property if and only if it looks like this:\n\nav(b f c)=avb f avc\n\nwhere a, b, and c are numbers, while v is one operator and f is the other. Here's how it works for multiplication when combined with addition:\n\na(b+c)=ab+ac\n\nFor example,\n\n5(3+1)=5(3) + 5(1)\n\nWhich, of course, is equal to 15+5, or 20.
44177
I'm sorry, but I don't think that you have actually asked a question here, or perhaps you have left off some very pertinent information. Unless you just wanted the above statement to be explained.\n\nIn this instance, the universe of discourse just means all the points contained in the set S = (p(x,y)), which is all the possible combinations of x = 1,2,3 and y = 1,2,3. For example, (1,3) and (2,2) and (3,1) would all be points contained in the universe of discourse for p(x,y).
785952
So you need to prove that each element (x,y) E br(xo, yo) has a neighbourhood contained in br(xo, yo),\n\nTake a point (xa,ya) E br(xo,yo) and let d = r - ||(xa,ya) - (xo,yo)||.  d is positive since ||(xa,ya) - (xo,yo)|| < r.\nThen the neighbourhood bd(xa,ya) satisfies the condition because for any point (x,y) in bd(xa,ya) we have\n||(x,y) - (xo,yo)||\n<= ||(x,y) - (xa,ya)|| + ||(xa,ya) - (xo,yo)|| (triangle inequality)\n< d + ||(xa,ya) - (xo,yo)|| (because (x,y) E bd(xa,ya))\n= r.\n\nIn plain English: If the point (xa,ya) has distance d from the boundary of the set, then a neighbouroud of (xa,ya) whose elements are all less than d away from (xa,ya) will be contained in the set as well.
617264
Since each term has at least x in it, divide out the x by distributive property:\n\nx(x^2 - 4xy + 4y^2)\n\nThen factor the rest by FOIL:\n\nx(x-2y)(x-2y)
819941
http://www.usconstitution.net/const.html#Article1
837082
Hint: Let C=X&#92;O and let f(x)=dist(x,C). The other way is trivial.
443950
Is the = really a minus?  If so, the GCF of each term is\n4a^2b, because the GCF of 4 and 8 is 4, and the GCF of variables has the lower of the exponents of each variable.\n\nDivide each term by this GCF (when you divide, subtract the exponents) and write the GCF outside () with the division answer inside (distributive property in reverse)\n\n4a^2b(b^2 - 2ac)
757825
There has been no development in this field, but you can check out the following result pertaining to set theory and group theory.\n\nThe power set P(x) of a set X, is a not a group under (U) union operation.\n\nProof : \nTo prove this we try to prove three properties \n1. Associativity \nFor A, B, C are in P(x) then \n(AUB)UC = A U (BUC)  [This is more or less obvious or simple to prove]\n\n2. Identity element\nThere is an an element(e) in P(x) such that for all A in P(x)\ne U A = A U e = A\nAnd, this element for power set is 'empty set' . [Sorry, but I dont have 'phi' symbol here.]\n\n3. Inverse element\nFor each A  in P(x) there exist another element B in P(x) such that\n(AUB) = (BUA) = e(empty set)\n\nThis property is not possible because you can not produce an empty set with union operation with any other set(irrespective wether its empty or not)\n\nSo, its not a group under Union.\n\n\nPlease noe that : The power set P(x) of a set X, is a not a group under intersection operation.
352597
well, since you're using 1, i'm pretty sure the property you're looking for the multiplication property of identity, which basically states that anything multiplied by one is itself.
189985
Use the distributive property of multiplication in reverse.  When you factor an expression, the goal is to get another expression that is in some way simpler (say because you can see the factors) and you don't want to change the value of the expression.\n\nAn example of the distributive property in action:\n3(3x+4) = 9x + 12\n\nTo apply it in reverse, you basically multiply the expression by:\n\n1 = factor/factor\n\nor:\n\n3/3 * (9x+12) = 3 * 1/3 * (9x+12) =...\n...             3 *(9/3*x+12/3)   =...\n...             3 *(3x+4)\n\nI hope this is what you mean by "factoring"\n\nExponentials and logarithms have factoring rules as well.
930651
Distributive property:\n8p+8+3p=-14\n\nCombine like terms:\n11p+8=-14\n\nSubtract 8 on both sides:\n11p=-22\np=-2\n\nCheck:\n8(-2+1)+3(-2)=-14\n8(-1)+3(-2)=-14\n-8-6=-14\n-14=-14
624409
Clearly false. Let K={i}.\n\nDid you mean for K to be a subfield? In that case, K contains the rationals and so is dense in R.
1042552
Here is my proof. Suppose P(x) is a polynomial in odd degree, for example 4x^5 + 3x^4 - 5x^3 + 2x^2 - 6x + 9. For large enough values of x, the term in x^5 will be so much larger than any of the other terms that it will swamp the smaller powers of x, and x^5 alone will determine the value of P. \n\nNow, if P is of odd degree and x is very large and positive, P will be positive; if x is very large and negative, then P will be negative. Therefore, somewhere between these two values of P there must be at least one point where P = 0. QED.
646611
I'm not cimpletely sure which axioms you are using, so I have to guess to some extent. One way is as follows:\nFrom 3x=9, add -9 to both sides to get\n3x+(-9)=9+(-9)=0\nwhere we have used the defining property of negatives.\nThis gives 3(x-3)=0 by the distributive law.\nNow, you either use the existence of multiplicative inverses to multiply by 3^(-1) or use the fact that xy=0 implies x=0 or y=0 (depending on your axioms) to get that\n3=0 or x-3=0 OR (x-3)=3^(-1)*3(x-3)=x-3=3^(-1)*0=0.\nIn either case, add 3 to both sides (since we know 3 is not 0) to get that x=0+3=3.
551184
its simple distributive law.\nwhat has happened is they have taken (a-2) common out of both the terms .\neg: cx-x = x(c-1)\nthe same thing has happened here with x=(a-2) and c=a^2\nso, in step 2, we have: a^2(a-2)-(a-2)\ntaking (a-2) common outside; you end up with (a-2)*((a^2)-1)\nwhihc is [3]\nthen as you know, (a^2)-1 is same as (a^2)-(1^2) and then we apply the algebraic identity (a^2)-(b^2)=(a+b)*(a-b) to get (a-2)(a+1)(a-1) which is [4]
1144405
1. multiplicative inverse for x is x^(-1) or 1/x\nx*x^(-1)=1\n\n\nexample: 1/5 is multiplicative inverse for 5 because\n5*[1/5]=1\n\n2. commutative law for addition is \na+b=b+a\n\nexample: 3+5 = 5+3\n(8=8)\n\ncommutative law for multiplication is ab=ba\n\nexample: 5*3=3*5\n(15=15)\n\n3. distributive property means a(b+c)=ab+ac\n\nexample: 5(1+3)=5*1+5*3=5+15=20.
932467
Distributive Property:\n6x-42-2x+8=13\n\nCombine like terms:\n4x-34=13\n\nAdd 34 on both sides:\n4x=47\nx=47/4\n\nCheck:\n6(47/4-7)-2(47/4-4)=13\n6(47/4-28/4)-2(47/4-16/4)=13\n6(19/4)-2(31/4)=13\n57/2-31/2=13\n26/2=13\n13=13