[新しいコレクション] (a-b-c)^3 formula expansion 269833-What is the expansion of (a+b+c)^3
In binomial theorem expansion, the binomial expression is most important in an algebraic equation which holds two different termsN C m = combination of n objects taken m at a timeA³ - 3a²b + 3ab² - b³-----= (a + b)(a + b)(a + b) = (a + b)(a² + ab + ab + b²) = (a + b)(a² + 2ab + b²) = a³ + 2a²b + ab² + a²b + 2ab² + b³ = a³ + 3a²b + 3ab² + b³
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What is the expansion of (a+b+c)^3
What is the expansion of (a+b+c)^3-Some Example of Binomial ExpansionA + b, a3 + b3, etc
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(c)All of the committees formed in (b) qualify except those in which Smith and Jones are3 3 + 3 × (3 2 × x) + 3 × (x 2 × 3) + x 3 (we are replacing a by 3 and b by x in the expansion of (a + b) 3 above) Generally It is, of course, often impractical to write out Pascal"s triangle every time, when all that we need to know are the entries on the nth lineX, y ∈ R;
Solution.We found the binomial coefficients to be 1 5 10 10 5 1The Expansion of (a + b)nBinomial formula for (a + b)3
Thermal expansion is large for gases, and relatively small, but not negligible, for liquids and solids⇒3 C0a3b0 +3 C1a2b1 +3 C2a1b2 +3 C3a0b3To get the value of (ab + bc + ac), we can use the formula or expansion of (a + b + c) 2
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∑ i = 0 n n C r x n − rWe want to get coefficient of a 3 b 2 c 4 d this implies that r 1 = 3, r 2 = 2, r 3 = 4, r 4 = 1, ∴ The coefficient of a 3 b 2 c 4 d is (10)!/(3!.2!.4) (-1) 2 (-1)-4 =Students, teachers, parents, and everyone can find solutions to their math problems instantly
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= a4 +4a3x+6a2x2 +4ax3 +x4Notice the "·" between the 3 and 6 to mean multiply, so 3·6 = 18:( a + b) n = a n + n C 1 a n − 1 b + n C 2 a n − 2 b 2 + ⋯ + n C m a n − m b m + ⋯ + b n
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( a + b) 2 = a 2 + 2 a b + b 2Expand (a − b) 5Using multinomial theorem, we have
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If a + b + c = 36 and a 2 + b 2 + c 2 = 676, then find the value of (ab + bc + ca)The lower index, in this case 4, is the exponent of b(a + b + c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ac (a + b + c) 2 = a 2 + b 2 + c 2 + 2(ab + bc + ac)
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So 3 (a+b) means to multiply 3 by (a+b) Here is an example of expanding, using variables a, b and c instead of numbers:This same number is also the coefficient of a 4 b 8, since 12 C 8 = 12 C 4Expand (a – b – c + d) 10 using multinomial theorem and by using coefficient property we can obtain the required result
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As →3 C0 =3 C3 = 1 and →3 C1 =3 C2 = 3A 3 +b 3 +c 3-3abc=(a+b) 3-3a 2 b-3ab 2 +c 3-3abc =(a+b+c) 3 -3(a+b) 2 c-3(a+b)c 2 -3ab(a+b+c) =(a+b+c) 3 -3(a+b)c(a+b+c)-3ab(a+b+c)=(a+b+c)(a 2 +b 2 +c 2 -ab-bc-ac)Here, a = x and b = 1
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Write the formula / expansion for (a + b + c) 2N C n − 1 xIn general we see that the coe cients of (a + x) n come from the n-th row of Pascal's riangleT , in which each term is the sum of the two terms just above it
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( a + b) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3The 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 5Linear thermal expansion is Δ L = α L Δ T , where Δ L is the change in length L, Δ T is the change in temperature, and α is the coefficient of linear expansion, which varies slightly with temperature
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So the answer is:Y r + n C r x n − r= (a - b)(a - b)(a - b) = (a - b)(a² - ab - ab + b²) = (a - b)(a² - 2ab + b²) = a³ - 2a²b + ab² - a²b + 2ab² - b³ = a³ - 3a²b + 3ab² - b³
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⇒3 C0x3 +3 C1x2 × 11 +3 C2x1 ×12 +3 C3 × 13Then the result will be
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