My first and naive impression is that the result is 0 but according to Salinas, Introduction to Statistical Physics that's $3x^{1/2}y O(x/y)^3$ I think Taylor expansion would do it The thingNow, we have the coefficients of the first five terms By the binomial formula, when the number of terms is even, then coefficients of each two terms that are at The Binomial Theorem is the method of expanding an expression which has been raised to any finite power A binomial Theorem is a powerful tool of expansion, which has application in Algebra, probability, etc Binomial Expression A binomial expression is an algebraic expression which contains two dissimilar terms Ex a b, a 3 b 3, etc
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How do you expand (x+y)^3
How do you expand (x+y)^3- 0 Follow 0 A K Daya Sir, added an answer, on 25/9/13 A K Daya Sir answered this x 3 y 3 = (x y) (x 2 xy y 2 ) this formula can be derived from (x y) 3 = x 3 y 3 3xy (x y) x 3 y 3 = (x y) 3 3xy (x y) x 3 y 3 = (x y) (x y) 2 3xy = (x y) x 2 y 2 2xy 3xy = (x y) (x 2 xy y 2 ) Was this answerKey Takeaways Key Points According to the theorem, it is possible to expand the power latex(x y)^n/latex into a sum involving terms of the form latexax^by^c/latex, where the exponents latexb/latex and latexc/latex are nonnegative integers with latexbc=n/latex, and the coefficient latexa/latex of each term is a specific positive integer depending on latexn/latex
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So, let's try to derive the identity x 3 y 3 using the identity for (x y) 3 Let's first try to understand this geometrically Let's join our cubes as shown above We know the algebraic expansion of (x y) 3 Rearranging the terms in the expansion, Substituting the values of x and y in the formula, Thus, we have factorized aYou can check the formulas of A plus B plus C Whole cube in three ways We are going to share the (abc)^3 algebra formulas for you as well as how to create (abc)^3 and proof we can write we know that what is the formula of need too write in simple form of multiplication Simplify the all Multiplication one by oneFind the product of two binomials Use the distributive property to multiply any two polynomials In the previous section you learned that the product A (2x y) expands to A (2x) A (y) Now consider the product (3x z) (2x y) Since (3x z) is in parentheses, we can treat it as a single factor and expand (3x z) (2x y) in the same
Answer by lenny460 (1073) ( Show Source ) You can put this solution on YOUR website! The simplify command finds the simplest form of an equation Simplifyexpr,assum does simplification using assumptions Expandexpr,patt leaves unexpanded any parts of expr that are free of the pattern patt ExpandAllexpr expands out all products and integer powers in ant part of exps ExpandAllexpr,patt avoids expanding parts of expr that do not contain terms matchingThe Binomial theorem tells us how to expand expressions of the form (ab)ⁿ, for example, (xy)⁷ The larger the power is, the harder it is to expand expressions like this directly But with the Binomial theorem, the process is relatively fast!
Expand (xy)^3 (x y)3 ( x y) 3 Use the Binomial Theorem x3 3x2y3xy2 y3 x 3 3 x 2 y 3 x y 2 y 3In mathematics, the cube of sum of two terms is expressed as the cube of binomial x y It is read as x plus y whole cube It is mainly used in mathematics as a formula for expanding cube of sum of any two terms in their terms (x y) 3 = x 3 y 3 3 x 2 y 3 x y 2Binomial Theorem to expand polynomials explained with examples and several practice problems and downloadable pdf worksheet (x y) 4 = x 4 4x 3 y 6x 2 y 2 4xy 3 y 4;
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View this answer We can expand x3−y3 x 3 − y 3 using the known identity (x−y)3 = x3−y3 −3xy(x−y) ( x − y) 3 = x 3 − y 3 − 3 x y ( x − y) This is done as follows $$\beginPolynomial Identities When we have a sum (difference) of two or three numbers to power of 2 or 3 and we need to remove the brackets we use polynomial identities (short multiplication formulas) (x y) 2 = x 2 2xy y 2 (x y) 2 = x 2 2xy y 2 Example 1 If x = 10, y = 5a (10 5a) 2 = 10 2 2·10·5a (5a) 2 = 100 100a 25a 2Example Expand 3 × (52) Answer It is now expanded We can also complete the calculation 3 × (52) = 3 × 5 3 × 2 = 15 6 = 21 In Algebra In Algebra putting two things next to each other usually means to multiply So 3(ab) means to multiply 3 by (ab) Here is an example of expanding, using variables a, b and c instead of numbers
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An outline of Isaac Newton's original discovery of the generalized binomial theorem Many thanks to Rob Thomasson, Skip Franklin, and Jay Gittings for their(xyz)^3 put xy = a (az)^3= a^3 z^3 3az ( az) = (xy)^3 z^3 3 a^2 z 3a z^2 = x^3y^3 z^3 3 x^2 y 3 x y^2 3(xy)^2 z 3(xy) z^2 =x^3 y^3 z^3 3 x compared with ( 2 x − x 2) 6 we can use the substitution y = x − x 2 and the result is (2) ( 2 x − x 2) 2 = 64 192 ( x − x 2) 240 ( x − x 2) 2 160 ( x − x 2) 3 ⋯ Since we only need an expansion with powers up to x 3 we don't need any terms ( x − x 2) n with n > 3 We also recall the binomial formulas ( a b) n
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(x y) 7 = x 7 7x 6 y 21x 5 y 2 35x 4 y 3 35x 3 y 4 21x 2 y 5 7xy 6 y 7 When the terms of the binomial have coefficient(s), be sure to apply the exponents to these coefficients Example Write out the expansion of (2 x 3 y ) 4There are various student are search formula of (ab)^3 and a^3b^3 Now I am going to explain everything below You can check and revert back if you like you can also check cube formula in algebra formula sheet a2 – b2 = (a – b)(a b) (ab)2 = a2 2ab b2 a2 b2 = (a – Expand the first two brackets (x −y)(x − y) = x2 −xy −xy y2 ⇒ x2 y2 − 2xy Multiply the result by the last two brackets (x2 y2 −2xy)(x − y) = x3 − x2y xy2 − y3 −2x2y 2xy2 ⇒ x3 −y3 − 3x2y 3xy2 Always expand each term in the bracket by all the other terms in the other brackets, but never multiply two or
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The following are algebraix expansion formulae of selected polynomials Square of summation (x y) 2 = x 2 2xy y 2 Square of difference (x y) 2 = x 2 2xy y 2 Difference of squares x 2 y 2 = (x y) (x y) Cube of summation (x y) 3 = x 3 3x 2 y 3xy 2 y 3 Summation of two cubes x 3 y 3 = (x y) (x 2 xy y 2) CubeThe calculator can also make logarithmic expansions of formula of the form ln ( a b) by giving the results in exact form thus to expand ln ( x 3), enter expand_log ( ln ( x 3)) , after calculation, the result is returned The calculator makes it possible to obtain the logarithmic expansionTheorem For any positive integer m and any nonnegative integer n, the multinomial formula describes how a sum with m terms expands when raised to an arbitrary power n ( ) = = (,, ,) =,where (,, ,) =!!!!is a multinomial coefficientThe sum is taken over all combinations of nonnegative integer indices k 1 through k m such that the sum of all k i is nThat is, for each term
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