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\( \hspace{15em} \ = \ {a}^{3} 3{a}^{2}b 3a{b}^2 {b}^{3} \) Of course this is a really tedious way for a very large index/power/exponent number Instead of doing it manually, we could use a formula called the Binomial Theorem which is shown belowBinomial Expansion Concepts & Theory If n is a positive integer, (a x) n can be expanded as (a x) n = a n n C 1 a n1 x n C 2 a n2 x 2 n C 3 a n3 x 3 x n Applying binomial on (a b) 2 Using general expansion equation a 2 c 1 (a) 21 (b) 1 b = a 2 2ab b 2Question 10 Find the ratio of the 5 th term from the beginning to the 5 th term from the end in the binomial expansion of 2 1/3 1/{2(3) 1/3} 10 Solution Question 11 Find the coefficient of a 3 b 2 c 4 d in the expansion of (abcd) 10 Solution

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(a+b)^3 binomial expansion-Binomial Expansion of (2i)^4;Ab is a binomial (the two terms are a and b) 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 b4

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Here you will see how to find a term or coefficient in a Binomial Expansion formula for positive integer exponents https//wwwyoutubecom/playlist?list=PL511 Pascal's triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 15 6 1 To expand (ab)n we look for the row starting with 1 and n 12 Example Let'sX, y ∈ R;

In binomial theorem expansion, the binomial expression is most important in an algebraic equation which holds two different terms Such as a b, a3 b3, etc Let's consider;2 See answers afmill01 afmill01 Pascal triangles gives us the coefficients for an expanded binomial of the form (ab)n,where n is now the row of the triangle shanmso2 shanmso2 The coefficients of the terms come from rows of(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 You realized that expanding a binomial expression by direct multiplication as shown above is quite cumbersome and inapplicable for larger exponents

Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with stepbystep explanations, just like a math tutorBinomial Expansion of (2i)^4;There is always a trick (don't forget that line, it appears in Math a lot) Observe that a − b a b a−b is the same as a ( − b) a (b) a(−b) Ahhhhhhh, so the Binomial Theorem still applies So then ( a − b) 3 = ( a ( − b)) 3 = a 3 ( 3 1) a 2 ( − b) ( 3 2) a ( − b) 2 ( − b) 3

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1 Binomial expansion We know that (ab)1 = ab (ab)2 = a2 2abb2 (ab)3 = a3 3a2b3ab2 b3 The question is (at this stage) what about (ab)n where n is any positive integer?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 miracleBinomial Expansion Examples 1 Expand (ab) 5 using binomial theorem Solution Here, the binomial expression is (ab) and n=5 So, using binomial theorem we have, 2 Find the middle term of the expansion (ax) 10 Solution Since, n=10(even) so the expansion has n1 = 11 terms Hence there is only one middle term which is

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Example 3 Expand the expression (ax b)3 The 4th row of Pascal's triangle is 1 3 3 1 So (ax b)3 = (ax)3 3(ax)2b = a 3x 3a2bx2 Notice that the powers of a and b in each term always add to give the power of the expansion This is always the case Example 4 Expand the expression (2x 3)3 Using the 4th row of pascals triangleThe expansion of math(ab)^{n}/math cannot be derived from the binomial formula for positive integral exponents, solely due to the absurdity of the term math\binom{n}{r}/math for any mathn < 0/math However, one can obtain this expansIn using the binomial formula, we let `a = 2x,` `b = 3` and `n = 4` Substituting in the binomial formula, we get `(2x3)^4` `=(2x)^44(2x)^3(3)` `(4(3))/(2!)(2x)^2(3)^2` `(4(3)(2))/(3!)(2x)(3)^3(3)^4` `=16x^496x^3216x^2216x81`

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X, y ∈ R;Using the Formula of Expansion of Binomial of Power 3 Examples Question 1 Find x 3 y 3, if x y = 5 and xy = 14 Solution x 3 y 3, if x y = 5 and xy = 14 (a 3 b 3) = (a b) 3 3ab(a b) (x 3 y 3) = (x y) 3 3xy(x y) By using the given values, we get (x 3 y 3) = 5 3 3(14)(5) = 125 210 = 335In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial According 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 positive integer depending on n and b

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About the Author Davneet Singh Davneet Singh is a graduate from Indian Institute of Technology, Kanpur He has been teaching from the past 9 years He provides courses for Maths and Science at TeachooExpand using the Binomial Theorem (2ab)^3 Use the binomial expansion theorem to find each term The binomial theorem states Expand the summation Simplify the exponents for each term of the expansion Simplify each term Tap for more steps Multiply by Apply the product rule to Raise to the power of Apply the product rule to(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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If we make x and y equal to 1 in the following (Binomial Expansion) 11 We find the sum of the coefficients 12 Another way to look at 11 is that we can select an item in 2 ways (an x or a y), and as there are n factors, we have, in all, 2 n possibilities Sum of Coefficients for p ItemsThe binomial expansion has multiple applications in Algebra and in Probability Theory For example, in Probability, the Binomial distribution is based upon the binomial theorem Indeed, consider a number \(0 \le p \le 1\) Then, \(p (1p) = 1\) and we can use the Binomial theoremBinomial Expansion Calculator is a free online tool that lets you solve the expansion of a binomial in the blink of an eye Just enter the input term in the below box and tap on the calculate button to attain the result in Binomial Expansion Ex Binomial Expansion of (2h)^3;

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A 3 − b 3 = (a − b) (a 2 b 2 ab) a 3 b 3 Decimal Expansion;The binomial theorem is used to describe the expansion in algebra for the powers of a binomial According to this theorem, it is possible to expand the polynomial \((x y)^n\) into a series of the sum involving terms of the form a \(x^b y^c\)The expansion of (ab)^3 is(ab)^3 (a b)^3 ( a b ) ^ 3 ( a b ) ^ 3 and this will go on for ever

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What are the coefficients for the binomial expansion of (a b)3?Binomial Expansion of (2x)^2Algebra Expand using the Binomial Theorem (2ab)^3 (2a − b)3 ( 2 a b) 3 Use the binomial expansion theorem to find each term The binomial theorem states (ab)n = n ∑ k=0nCk⋅(an−kbk) ( a b) n = ∑ k = 0 n ⁡ n C k ⋅ ( a n k b k) 3 ∑ k=0 3!

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Binomial Expansion Calculator The calculator will find the binomial expansion of the givenIt is a powerful tool for the expansion of the equation which has a vast use in Algebra, probability, etc Binomial Theorem Expansion In binomial theorem expansion, the binomial expression is most important in an algebraic equation which holds two different terms Such as a b, a 3 b 3, etc Let's consider;Binomial Expansion Calculator is a free online tool that lets you solve the expansion of a binomial in the blink of an eye Just enter the input term in the below box and tap on the calculate button to attain the result in Binomial Expansion Ex Binomial Expansion of (2h)^3;

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(ab) 3 = 3 C 0 a 3 3 C 1 a 2 b 3 C 2 ab 2 3 C 3 b 3 So from the above pattern, the expansion of (a b) n becomes as follows (a b) n = n C 0 a n n C 1 a n1 b 1 n C 2 a n2 b 2 n C r a nr b r n C n b n The above statement is called Binomial Theorem Binomial Theorem FormulaExpansion of Binomial Expression In order to expand binomial expression, we use repeated multiplication For example, (ab) 2 = (ab) (ab) = a (ab) b (ab) = a 2 2ab b 2 (mn) 3 = (mn) (mn) 2 = (mn) (m 2 2mn n 2 ) = m (m2 2mn n2) n (m2 2mn n2) = m 3 3m 2 n 3mn 2 n 31 3 3 1

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Solution We found the binomial coefficients to be 1 5 10 10 5 1 The difference with ( a − b) is that the signs of the terms will alternate ( a − b) 5 = a5 − 5 a4b 10 a3 b 2 − 10 a2b3 5 ab4 − b5 For, a − b = a (− b ), therefore each term will have the form a5 − k (− b) kBinomial Coefficient In (a b)2 = 1a2 2ab 1b2 1, 2 and 1 are known as the binomial coefficients of a2, ab and b2 respectively Likewise, in (a b)3 = 1a3 3a2b 3ab2 1b3 1, 3, 3 and 1 are known as the binomial coefficients of a3, a2b, ab2 and b3 respectively Generally, in the expansion of (x y)nBinomial Theorem Calculator online with solution and steps Detailed step by step solutions to your Binomial Theorem problems online with our math solver and calculator Solved exercises of Binomial Theorem

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Pascal's triangle can be used to find the coefficient of binomial expansion (a b) 0 1 (a b) 1 1 1 (a b) 2 1 2 1 (a b) 3 1 3 3 1 (a b) 4 1 4 6 4 1 (a b) 5 1 5 10 10 5 1 (a b) 6 1 6 15 15 6 1 (a b) 7 1 7For any binomial (a b) and any natural number n, The binomial theorem can be proved by mathematical induction (See Exercise 63) This form shows why is called a binomial coefficient Example 3 Expand (x 2 2y) 5 Solution We have (a b) n,where a = x 2, b = 2y, and n = 5 Then using the binomial theorem, we haveApplying Binomial on (a b) 3 a 30 3 c 1 a 31 b 1 3 c 2 a 32 b 2 b 30 = a 3 3a 2 b 3ab 2 b 3 And if in this case if particular term say 'r' is asked, then instead of applying the whole expansion, the following direct formula can be applied to find the 'r'th term

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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 MathematicsN ∈ N Then the result will be ∑ i = 0 n n C r x n − r y r n C r x n − r y r n C n − 1 x y n − 1 n C n y nTo find this we use the row beginning 1,4, and can immediately write down the expansion (ab)4 = a4 4a3b6a2b2 4ab3 b4 We can apply the same procedure to expand any binomial expression, even when the quantities a and b are more complicated Consider the following examples Example Suppose we want to expand (2xy)3

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Binomial Expansion of (2x)^2Evaluate 136 using the Binomial Theorem Write 13 as a binomial (1 __) Complete the expansion of (a b)6 with a = 1 and b = 03 (1 03)6 = (1)6 6(1)5(03) 15(1)4(03)2(1)3(03)315(1)2(03)4 6(1)(03)5 (03)6 After evaluating the powers, the expression reduces to =This is equal to a to the 4th plus, plus 4, plus 4a to the 3rd, a to the 3rd b plus, plus 6, plus 6a squared b squared, a squared b squared, plus, plus, plus 4, I think you see a pattern here, plus 4a times b to the 3rd power plus b to the 4th power, plus b to the 4th power

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A b)2 = 1a2 2ab 1b2 (a b)3 = 1a3 3a2b 3ab2 1b3 How are binomial expansions related to Pascal's triangle?Learn how to find the coefficient of a specific term when using the Binomial Expansion Theorem in this free math tutorial by Mario's Math Tutoring010 Examp⋅(2a)3−k ⋅(−b)k ∑ k = 0 3 ⁡

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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#Binomial Expansion Calculator The 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)`If a binomial expression (x y) n is to be expanded, a binomial expansion formula can be used to express this in terms of the simpler expressions of the form ax by c in which 'b' and 'c' are non negative integers The value of 'a' completely depends on the value of 'n' and 'b'

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Binomial Expansion is one of the methods used to expand the binomials with powers in algebraic expressions In algebra, a binomial is an algebraic expression with exactly two terms (the prefix 'bi' refers to the number 2) If a binomial expression (x y)n is to be expanded, a binomial expansion formula can be used to express this in terms of the simpler expressions of the form ax by c in which 'b' and 'c' are non negative integersThe Binomial Expansion Powers of a b So, we now have 3) (ba 2) (ba CoefficientsExpression 1 2 1 1 3 3 1 4) (ba 1 4 6 4 1 The 1st and last numbers are always 1 Each number in a row can be found by adding the 2 coefficients above it 15In the binomial expansion of (a b) n, the coefficient of fourth and thirteenth terms are equal to each other, then the value of n is (a) 10 (b) 15 (c) (d) 25 Answer Answer (b) 15 Hint Given, in the binomial expansion of (a b) n, the coefficient of fourth and thirteenth terms are equal to each other ⇒ n C 3 = n C 12 This is possible

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(a b) 12 = a 12 12a 11 b 66a 10 b 2 2a 9 b 3 495a 8 b 4 792a 7 b 5 924a 6 b 6 792a 5 b 7 495a 4 b 8 2a 3 b 9 66a 2 b 10 12ab 11 b 12 and so on In case of the expansion of smaller powers, it is difficult to calculate the coefficients of the binomial expansion using the same statement

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