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A commonly misunderstood topic in precalculus is the expansion of binomials In this video we take a look at what the terminology means, make sense of theMultiply both sides of the equation by 3 x y, the least common multiple of x y, 3 Use the distributive property to multiply 3 by 2x^ {2}5y^ {2} Use the distributive property to multiply 3 by 2 x 2 − 5 y 2 Subtract xy from both sides Subtract x y from both sidesBinomial Expansions Binomial Expansions Notice that (x y) 0 = 1 (x y) 2 = x 2 2xy y 2 (x y) 3 = x 3 3x 3 y 3xy 2 y 3 (x y) 4 = x 4 4x 3 y 6x 2 y 2 4xy 3 y 4 Notice that the powers are descending in x and ascending in yAlthough FOILing is one way to solve these problems, there is a much easier way

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(x y)^3 expansion-(x y) 1 6? Explanation (x −y)3 = (x − y)(x −y)(x −y) 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

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X^3 y^3 z^3 3x^2y 3xy^2 3x^2z 3z^2x 3y^2z 3z^2y 6xyz Lennox Obuong Algebra Student Email obuong3@aolcomOur online expert tutors can answer this problem Get stepbystep solutions from expert tutors as fast as 1530 minutes Your first 5 questions are on us!Created by T Madas Created by T Madas Question 3 (**) y x= e sin32x a) Use standard results to find the series expansion of y, up and including the term in x4 b) Hence find an approximate value for 01 2 0 e sin3x x dx FP2M , e sin3 3 6 52 2 3 4 53 ( ) 2 x x x x x x O x

 Hi Heureka, I am wondering if it can still be done easily if it is made more complicated ?In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomialAccording to the theorem, it is possible to expand the polynomial (x y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b c = n, and the coefficient a of each term is a specific positiveExpansion of ( )2 3− x 10 b) Use the first three terms in the binomial expansion of ( )2 3− x 10, with a suitable value for x, to find an approximation for 197 10 c) Use the answer of part (b) to estimate, correct to 2 significant figures, the value of 394 10

In the denominator for each term in the infinite sum History The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility;Jee mains 1 vote 1 answer Find the coefficient of the term x^9 in the expansion of (2x4)^9 using binomial therorem asked in Mathematics by Abdulazeez John Bami (122 points) binomialAnswer (1 of 15) Mentally examine the expansion of (xyz)^3 and realize that each term of the expansion must be of degree three and that because xyz is cyclic all possible such terms must appear Those types of terms can be represented by x^3, x^2y and xyz If x^3 appears, so must y^3 and z^3

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The expansion is given by the following formula $$$ \left(a b\right)^{n} = \sum_{k=0}^{n} {\binom{n}{k}} a^{n k} b^{k} $$$, where $$$ {\binom{n}{k}} = \frac{nFirst week only $499!We know the algebraic expansion of (x y) 3 Rearranging the terms in the expansion, we will get our identity for x 3 y 3 Thus, we have verified our identity mathematically Again, if we replace x with − y in the expression, we have This identity can also be expressed as, This is the required standard algebraic identity In the expression, if we replace y with (− y), we will get the

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RD Sharma Class 10 Solutions; In the expansion of (2x y ) 3 (2x y) 3, the coefficient of x 2 y is This question was previously asked in SSC Matric Level Previous Paper (Held on Shift 1) Download PDF Attempt Online View all SSC Selection Post Papers > 24;Theorem 1 (The Trinomial Theorem) If , , , and are nonnegative integer such that then the expansion of the trinomial is given by Proof Let Consider the expansion of the trinomial For each factor we choose to distribute through one of the three variables , or many times we choose to expand through , many times we choose to expand

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0 (0) (0) (0) Choose An Option That Best Describes Your Problem Answer not in Detail Incomplete Answer Answer IncorrectStart your free trial In partnership with You are being redirected to Course Hero I want to submit the same problem to Course Hero Cancel(x y) 5 = x 5 5x 4 y 10x 3 y 2 10x 2 y 3 5xy 4 y 5 Look familiar?

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(x y) 3 = x 3 3x 2 y 3xy 2 y 3 (x y) 4 = x 4 4x 3 y 6x 2 y 2 4xy 3 y 4;X6 y3 (2)3The expansion of (ax)2 is (ax)2 = a2 2axx2 Hence, (ax)3 = (ax)(ax)2 = (ax)(a2 2axx2) = a3 (12)a 2x(21)ax x 3= a3 3a2x3ax2 x urther,F (ax)4 = (ax)(ax)4 = (ax)(a3 3a2x3ax2 x3) = a4 (13)a3x(33)a2x2 (31)ax3 x4 = a4 4a3x6a2x2 4ax3 x4 In general we see that the coe cients of (a x)n come from the nth row of Pascal's riangleT , in

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Solve the following by Taylor series Expansion 1 = x y' y=1;What is the coefficient of x 2 * y 3 in the expansion of 2x y 5?Thank you taylorexpansion Share Cite Follow edited Mar 9 '16 at 024 Michael Hardy 257k 28 28 gold badges 258 258 silver badges 549 549 bronze badges asked Mar 9 '16 at 000 Patrick Patrick 219 2 2 silver badges 9

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See Answer Check out a sample Q&A here Want to see this answer and more?Polynomial 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 23 Utilize the Binomial Expansion Calculator and enter your input term in the input field ie, ( 2 x − y) 3 & press the calculate button to get the result ie, 8 x 3 − 12 x 2 y 6 x y 2 − y 3 along with a detailed solution in a fraction of seconds Ex (x1)^2 (or) (x7)^7 (or) (x3)^4

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Answer (Detailed Solution Below) Option 1 24 Free Tests View all Free tests > Free GK Chapter Test 1 Ancient HistoryArrow_forward Question View transcribed image text fullscreen Expand check_circle Expert Answer Want to see the stepbystep answer?The above expansion holds because the derivative of e x with respect to x is also e x, and e 0 equals 1 This leaves the terms (x − 0) n in the numerator and n!

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Chapter 5 Expansion formulae Practice Set 52 Q 2 Page 25 Advertisement Remove all ads Question Bank with Solutions Maharashtra Board Question Bank with Solutions (Official) Textbook Solutions Balbharati Solutions (Maharashtra) Samacheer Kalvi Solutions (Tamil Nadu) NCERT Solutions; Example 7 Find the coefficient of x6y3 in the expansion of (x 2y)9 We know that General term of expansion (a b)n is Tr1 = nCr an–r br For (x 2y)9, Putting n = 9 , a = x , b = 2y Tr 1 = 9Cr (x)9 – r (2y)r = 9Cr (x)9 – r (y)r (2)r We need to find coefficient of x6 y3 Comparing yr = y3 r = 3 Putting r = 3 in (1) T31 = 9C3 x9 – 3 y3 (2)3 = 9!/3!(9 −3 )!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 3

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The coefficient of x2y3 in the expansion of (1xy) is !For example, consider the following expansion latex\displaystyle {(xy)}^{4}={x}^{4}4{x}^{3}{y}6{x}^{2}{y}^{2}4x{y}^{3}{y}^{4}/latex Any coefficient latexa/latex in a term latexax^by^c/latex of the expanded version is known as a binomial coefficient The binomial coefficient also arises in combinatorics, where it gives the$3x^{1/2}y O(x/y)^3$ I think Taylor expansion would do it The thing is, I don't really know around what point I should do it Could anyone help here?

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Eg Find the coefficient of x^3 y^3 z^2 in the expansion of (3x5y7z)^8Suppose we want to expand (2xy)3 We pick the coefficients in the expansion from the relevant row of Pascal's triangle (1,3,3,1) As we move through the terms in the expansion from left to right we remember to decrease the power of 2x and increase the power of y So, (2xy)3 = 1(2x)3 3(2x)2y 3(2x)1y2 1y3 = 8x3 12x 2y 6xy y3 Example Suppose we want to expand (1p)4 Using Binomial Expansion, (x y)³ = 3C0 * x³ 3C1 * x²y 3C2 * xy² 3C3 * y³ Therefore the coefficient of xy² is 3C2 = 3 sikringbp and 1 more users found this answer helpful heart outlined Thanks 1 star outlined star outlined

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The result was Zeno'sThank you Best wishes PG M MrAl WellKnown Member Most Helpful Member #4 Hi, Well, you dont really 'expand' that in the same way you dont expand sqrt(xFactor x^3y^3 x3 − y3 x 3 y 3 Since both terms are perfect cubes, factor using the difference of cubes formula, a3 −b3 = (a−b)(a2 abb2) a 3 b 3 = ( a b) ( a 2 a b b 2) where a = x a = x and b = y b = y (x−y)(x2 xyy2) ( x y) ( x 2 x y y 2)

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Really Haha yes But its just for fun I am reallyThis has both positive and negative terms, so it can be compared with the expansion of (x − y) 3 The terms of polynomials are rearranged Then terms that are perfect cubes are identified Comparing the polynomial with the identity we have, x = 2 a & y = 3 b Using the values of x & y, other terms of the polynomials are written as shown Since x 3 − 3 x 2 y 3 x y 2 − y 3 = (x − y) 3N= 3 di 2 fix) = 3x' 4x 2x 1 at x =0 and n= 3 close Start your trial now!

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Answer The coefficient of x²y³ is 40 What is the x 2 y 3 coefficient when expanding X − y 5?Find the coefficient of x6y3 in the expansion of (x 2y)9 (A) 674 (B) 670 676 (D) 672 Check Answer and Solution for above question from MathemaWhat is the coefficient of X 2y 3 in the expansion of 1 x/y ?

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Find the coefficient of x 6 y 3 in the expansion of (x 2 y) 9 Solution Solution (r 1) t h term in (x 2 y) 9 = C (9, r) x n − r (2 y) r = C (9, r) 2 r (x 9 − r y r) If we put, r = 3 Then, 4 t h term , T 4 = C (9, 3) ⋅ 2 3 (x 6 y 3) ∴ Coefficient of (x 6 y 3) = C (9, 3) ⋅ 2 3 = 8 ∗ 3 ∗ 2 ∗ 1 9 ∗ 8 ∗ 7 = 672 516 150 Connecting you to a tutor in 60 seconds GetIt is known that, \\left( a b \right)^3 = a^3 b^3 3 a^2 b 3a b^2 ; The first one LATEX\sqrt{y^63 x^2 y^43 x^4 y^2x^6}/LATEX By the way, do you have any clue that how one can expand the expressions involving fractional exponents such as {x^2y^2}^(1/2) on the calculators such TI?

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 Illustration 1 Expand (x/3 2/y) 4 Sol Illustration 2 (√2 1) 5 (√2 − 1) 5 Sol We have (x y) 5 (x – y) 5 = 25C 0 x 5 5C 2 x 3 y 2 5C 4 xy 4 = 2(x 5 10 x 3 y 2 5xy 4) Now (√2 1) 5 (√2 − 1) 5 = 2(√2) 5 10(√2) 3 (1) 2 5(√2)(1) 4 =58√2 Binomial Expansion Important points to remember The total number of terms in the expansion of (xy) n are Find the coefficient of the term x^6y^3 in the expansion of (x2y)^9 asked in Mathematics by Abdulazeez John Bami (122 points) binomial theorem; The Expansion Of 3x 2y 3 Is The expansion of (3x 2y) 3 is The given expression is of the form \(\begin{array}{l} (ab)^{3}=a^{3}b^{3}3 a^{2} b3 a b^{2} \\ (3 x2 y)^{3}=27 x^{3}8 y^{3}54 x^{2} y36 x y^{2} \end{array}\) Was this answer helpful?

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The coefficients of each expansion are the entries in Row n of Pascal's Triangle Thus, the coefficient of each term r of the expansion of (x y) n is given by C(n, r 1) The exponents of x descend, starting with n, and the exponents of y ascend, starting with 0, so the r th term of the expansion of (x y) 2The coefficient of x12 in the expansion of(x(y/x3))is (A) C8 (B) C8y8 C12 (D) C12 y12 Check Answer and Solution for above questi Check Answer and Solution for above questi TardigradeBinomial Theorem Formula The generalized formula for the pattern above is known as the binomial theorem Practice Problems on the Binomial Theorem Problem 1 Use the formula for the binomial theorem to determine the fourth term in the expansion (y − 1) 7 Show Answer Problem 2 Make use of

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Although the formula above is only applicable for binomials raised to an integer power, a similar strategy can be applied to find the coefficients of any linear polynomial raised to an integer power Submit your answer What is the coefficient of the x 2 y 2 zCalculate the expansion of $(xyz)^n$ Ask Question Asked 9 years, 10 months ago Active 3 years, 10 months ago Viewed 12k times 3 $\begingroup$ The question that I have to solve is an answer on the question "How many terms are in the expansion?" Depending on how you define "term" you can become two different formulas to calculate the terms in the expansion of $(xyFind the expansion of the following (xy2)^3 2 See answers 3 x y 2 Always expand each term in the bracket by all the other terms in the other brackets, but never multiply two or more terms in the same bracket Crystall91 Crystall91 (xy2)^3 => x^3y^32^3 (xy2)(xy2y2x) =>x^3y^38 (xy2)(xy2y2x) Cheers!

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X y is a binomial in which x and y are two terms In 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 2Free expand & simplify calculator Expand and simplify equations stepbystepRD Sharma Class 9 Solutions;

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1 Answer The coefficient of x3y2 in (x−3y)5 is 90\left( a b \right)^3 = a^3 b^3 3 a^2 b 3a b^2\ \\ \left( 5x 7y \right)^3An outline of Isaac Newton's original discovery of the generalized binomial theorem Many thanks to Rob Thomasson, Skip Franklin, and Jay Gittings for their

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