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Let us multiply ab by itself using Polynomial Multiplication (ab) (ab) = a2 2ab b2 Now take that result and multiply by ab again (a 2 2ab b 2 ) (ab) = a3 3a2b 3ab2 b3 And again (a 3 3a 2 b 3ab 2 b 3 ) (ab) = a4 4a3b 6a2b2 4ab3 b44 Binomial Expansions 41 Pascal's riTangle The 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'sWhat are the coefficients for the binomial expansion of (a b)3?

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(a+b+c)^3 expansion-Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with stepbystep explanations, just like a math tutorIn this video we will learn expansion of (ab)³, also solving practice set 53Expansion Formulae Part 1https//youtube/OPjWeOjHQ3sExpansion Formulae P

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Write the formula / expansion for (a b)3 (a b) 3 = a 3 3a 2 b 3ab 2 b 3 Substitute x for a and 1 for b (x 1) 3 = x 3 3 (x 2 ) (1) 3 (x) (1 2) 1 3 (x 1) 3 = x 3 3x 2 3 (x) (1) 1 (x 1) 3 = x 3 3x 2 3x 1 So, the expansion of (x 1)3 is x 3 3x 2 3x 1In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial According to the theorem, it is possible to expand the polynomial n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b c = n, and the coefficient a of each term is a specific positive integer depending on n and b For example, 4 = x 4 4 x 3 y 6 x 2 y 2 4 x y 3 y 4 {\displaystyle ^{4}=x^{4}4x^{3}y6x^{2}y^{2}4xy^{3}y^{4}} TheExpansion of (a b) 3 ∴ (a b) 3 = (a b)(a b)(a b) = (a b)(a b) 2 = (a b)(a 2 2ab b 2) = a(a 2 2ab b 2) b(a 2 2ab b 2) = a 3 2a 2 b ab 2 a 2 b 2ab 2 b 3 = a 3 3a 2 b 3ab 2 b 3 ∴ (a b) 3 = a 3 3a 2 b 3ab 2 b 3

A few simple expansion of (a b) to different powers (a b)2 = a2 2ab b2 (a b)3 = a3 3a2b 3ab2 b3 (a b)4 = a4 4a3b 6a2b2 4ab3 b4 You may begin to see a pattern in the coefficients They come from Pascal's Triangle 1 1 1 1 2 1 (a b)2#_3C_3 = (3!)/((33)!(3!)) = (3!)/(0!*3!) = 1# Note #(ab)^3 = (a (b))^3# Substitute into the Binomial expansion formula, let #x = a# and #y = b# #(ab)^3 = a^3 3a^2(b)^1 3a(b)^2(b)^3# #= a^3 3a^2b 3ab^2 b^3#So (a b)¹ = a b (a b)² = a² 2ab b² (a b)³ = a³ 3a²b 3b²a b³ You should notice that the coefficients of (the numbers before) a and b are 1 1 1 2 1 1 3 3 1 If you continued expanding the brackets for higher powers, you would find that the sequence continues 1 4 6 4 1

If we want to expand (ab)3 we select the coefficients from the row of the triangle beginning 1,3 these are 1,3,3,1 We can immediately write down the expansion by remembering that for each new term we decrease the power of a, this time starting with 3, and increase the power of b So (ab) 3= 1a 3a2b3ab2 1b3 which we would normally write as justWe can do this if you know the Binomial Theorem But if you don't know what it is or how to use it you can do the expansion on your own (ab)^3 = (ab)(ab)(ab) This becomes (ab)(ab)^2 =O 1, 4, 6, 4, 1 O 1,1 O 1, 2,1 O 1,3,3, 1 2 See answers esbeydeeromero esbeydeeromero The fourth option I got that last time happy244 happy244 Answer D Stepbystep explanation New questions in Mathematics

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MATHEMATICAL FORMULAE Algebra MATHEMATICAL FORMULAE Algebra 1 (ab)2=a22abb2;a2b2=(ab)2−2ab 2 (a−b)2=a2−2abb;a2b2=(a−b)22ab 3 (abc)2=a2b2c22(abbcca) 4 (ab)3=a3b33ab(ab);a3b3=(ab)−3ab(ab) 5Ex a b, a 3 b 3, etc Binomial Theorem Let n ∈ N,x,y,∈ R then n Σ r=0 nC r x n – r · y r nC r x n – r · y r nC n1 x · y n – 1 nC n · y n ie(x y) n = n Σ r=0 nC r x n – r · y r where, Illustration 1 Expand (x/3 2/y) 4 Sol Illustration 2 (√2 1) 5 (√2 − 1) 5 Sol We haveIf we want to expand (ab)3 we select the coefficients from the row of the triangle beginning 1,3 these are 1,3,3,1 We can immediately write down the expansion by remembering that for each new term we decrease the power of a, this time starting with 3, and increase the power of b So (ab) 3= 1a 3a2b3ab2 1b3 which we would normally write as just

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(a b) 0 = 1 (a b) 1 = a b (a b) 2 = a 2 2ab b 2 (a b) 3 = a 3 3a 2 b 3ab 2 b 3 (a b) 4 = a 4 4a 3 b 6a 2 b 2 4ab 3 b 4 (a b) 5 = a 5 5a 4 b 10a 3 b 2 10a 2 b 3 5ab 4 b 5 Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions Pascal's TriangleExercise 3 Expand the following expression, writing your answer in its simplest form Be careful of notation and do not use spaces in your answer ( x ) 2 = x 2 xWatch the Construction Specialties videos to learn more about expansion joint cover system testing and installation 051 See all Better Solutions We understand that every project has its own special movement criteria, and that one of our standard designs may not be right for your project We have a team of technical experts that can work

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Es Factores de expansión;The expansion will increase capacity, ease major congestion, and enhance travel time reliability The construction contract, owner's costs and contingency, brings the project's total budget to $38 billion, making it one of the largest infrastructure projects in the country A map of the project location is available here She's Big She'sTo find an expansion for (a b) 8, we complete two more rows of Pascal's triangle Thus the expansion of is (a b) 8 = a 8 8a 7 b 28a 6 b 2 56a 5 b 3 70a 4 b 4 56a 3 b 5 28a 2 b 6 8ab 7 b 8 We can generalize our results as follows The Binomial Theorem Using Pascal's Triangle For any binomial a b and any natural number n,

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We can write \((abc)^3 = (abc)(abc)(abc) \) \(=>(abc)^3 = (abc)^2 (abc) \) we know that what is the formula of \( (abc)^2 \) \(=>(abc)^3 = (a^2b^2c^2 2ab 2bc 2ca) (abc) \)To get `tan(x)sec^3(x)`, use parentheses tan(x)sec^3(x) From the table below, you can notice that sech is not supported, but you can still enter it using the identity `sech(x)=1/cosh(x)` If you get an error, doublecheck your expression, add parentheses and multiplication signs where needed, and consult the table below(abc) 3 a 3 b 3 c 3 We can choose three "a"'s for the cube in one way C(3,3)=1, or we can choose an a from the first factor and one from the second and one from the third, being the only way to make a3 The coefficient of the cubes is therefore 1 (It's the same for a, b and c, of course) 3a 2 b3a 2 c Next, we consider the a 2 terms We

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Westward Expansion (Manifest Destiny) 25k plays Qs Westward Expansion 17k plays 10 Qs Mexico Geography k plays Quiz not found!Sponsored Links Search the Engineering ToolBox search is the most efficient way to navigate the Engineering ToolBox!The difference with (a − b) is that the signs of the terms will alternate (a − b) 5 = a 5 − 5a 4 b 10a 3 b 2 − 10a 2 b 3 5ab 4 − b 5 For, a − b = a (−b), therefore each term will have the form a 5 − k (−b) k When k is even, (−b) k will be positive But when k is odd, (−b) k will be negative Each odd power of b will have a minus sign Example 3

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Expansion of (a b) 3 (a b) 3 = (a b)(a b)(a b) = (a b)(a b) 2 (a b) 3 = (a b)(a 2 2ab b 2) (a b) 3 = a(a 2 2ab b 2) b(a 2 2ab b 2) (a b) 3 = a 3 2a 2 b ab 2 ba 2 2ab 2 b 3 (a b) 3 = a 3 3a 2 b 3ab 2 b 3 ∴ (a b) 3 = a 3 3a 2 b 3ab 2 b 3Linear thermal expansion coefficient is defined as material's fractional change in length divided by the change in temperature Coefficient of linear thermal expansion is designated by the symbol α (alpha) The SI unit of thermal expansion coefficient is (°C)1 and USWe can do this if you know the Binomial Theorem But if you don't know what it is or how to use it you can do the expansion on your own (ab)^3 = (ab)(ab)(ab) This becomes (ab)(ab)^2 =

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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 And here is another example involving some numbers Notice the "·" between the 3 and 6 to mean multiply, so 3·6 = 18Cube Formulas (a b) 3 = a 3 b 3 3ab (a b) (a − b) 3 = a 3 b 3 3ab (a b) a 3 − b 3 = (a − b) (a 2 b 2 ab) a 3 b 3 = (a b) (a 2 b 2 − ab) (a b c) 3 = a 3 b 3 c 3 3 (a b) (b c) (c a) a 3 b 3 c 3 − 3abc = (a b c) (a 2 b 2 c 2 − ab − bc − ac) If (a b c) = 0,(abc) 3 a 3 b 3 c 3 We can choose three "a"'s for the cube in one way C(3,3)=1, or we can choose an a from the first factor and one from the second and one from the third, being the only way to make a3 The coefficient of the cubes is therefore 1 (It's the same for a, b and c, of course) 3a 2 b3a 2 c Next, we consider the a 2 terms We can choose two a's from 3 factors in C(3,2) ways=3

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Exercise 3 Expand the following expression, writing your answer in its simplest form Be careful of notation and do not use spaces in your answer ( x ) 2 = x 2 xIn mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials The expansion is given by The expansion is given by ( a b c ) n = ∑ i j k = n i , j , k ( n i , j , k ) a i b j c k , {\displaystyle (abc)^{n}=\sum _{\stackrel {i,j,k}{ijk=n}}{n \choose i,j,k}\,a^{i}\,b^{\;\!j}\;\!c^{k},}The expansion of (ab)^3 is(ab)^3 (a b)^3 ( a b ) ^ 3 ( a b ) ^ 3 and this will go on for ever Just write the expression on your page and see the miracle

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(a b) 3 = a 3 3 a 2 b 3 a b 2 b 3 (22) a 1/2 a 1/2 = a (23) a 1/3 a 1/3 a 1/3 = a Factors expansion;Return Policy View Return Policy $BACK TO EDMODO Menu Find a quiz All quizzes All quizzes My quizzes Reports Create a new quiz 0 Join a game Log in Sign up View profile Have an account?

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Expand the following product (3 x 1) (2 x 4) `(3x1)(2x4)` returns `3*x*2*x3*x*42*x4` Expand this algebraic expression `(x2)^3` returns `2^33*x*2^23*2*x^2x^3` Note that the result is not returned as the simplest expression in order to be able to follow the steps of calculations To simplify the results, simply use the reduce functionThe Senate voted 263 in favor of the effort to rework the expansion, which currently covers about 300,000 people in the state The measure now heads to the HouseMATHEMATICAL FORMULAE Algebra MATHEMATICAL FORMULAE Algebra 1 (ab)2=a22abb2;a2b2=(ab)2−2ab 2 (a−b)2=a2−2abb;a2b2=(a−b)22ab 3 (abc)2=a2b2c22(abbcca) 4 (ab)3=a3b33ab(ab);a3b3=(ab)−3ab(ab) 5

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Let us multiply ab by itself using Polynomial Multiplication (ab) (ab) = a2 2ab b2 Now take that result and multiply by ab again (a 2 2ab b 2 ) (ab) = a3 3a2b 3ab2 b3 And again (a 3 3a 2 b 3ab 2 b 3 ) (ab) = a4 4a3b 6a2b2 4ab3 b4Solution (2x 3)3 is in the form of (a b)3 Comparing (a b)3 and (2x 3)3, we get a = 2x b = 3 Write the formula / expansion for (a b)3 (a b) 3 = a 3 3a 2 b 3ab 2 b 3 Substitute 2x for a and 3 for b (2x 3) 3 = (2x) 3 3 (2x) 2 (3) 3 (2x) (3 2) 3 3The calculator will find the binomial expansion of the given expression, with steps shown Show Instructions In general, you can skip the multiplication sign, so `5x` is equivalent to `5*x` In general, you can skip parentheses, but be very careful e^3x is `e^3x`, and e^(3x) is `e^(3x)`

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R/BellyExpansion A Subreddit Dedicated To Female Belly Expansion Press J to jump to the feed Press question mark to learn the rest of the keyboard shortcuts3rd term = 0004 4th term ≈ 0 Rashad's Response There are 5 1 = 6 terms in the binomial expansion of (1−002)5, and since the 4th term is approximately 0, the 5th and 6th terms are also approximately 0 So, approximate the value of 0985 by adding the first three terms 1 (01) 0004 = 0904Will be right back Thank you for your patience Our engineers are working quickly to resolve the issue

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Log in now Create a new quizLet's expand (ab)3 The row in Pascal's triangle starting with 1 and 3 is 1 3 3 1 Therefore the expansion of (ab)3 is (ab)3 = a3 3a2b3ab2 b3 13 Example Let's expand (ab)6 The row starting with 1 and 6 in Pascal's triangle is the row 1 6 15 15 6 1 This means that the expansion of (ab)6 is (ab)6 = a6 6a5b15a4b2 a3b3 15a2b4 6ab5 b6 2When I raise it to the third power, the coefficients are 1, 3, 3, 1 When I raise it to the fourth power the coefficients are 1, 4, 6, 4, 1 and when I raise it to the fifth power which is the one we care about, the coeffiencients are going to be 1, 5, 10, 5 or sorry 10, 10, 5, and 1

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