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- The binomial theorem formula is (a+b) n = ∑ nr=0n C r a n-r b r, where n is a positive integer and a, b are real numbers, and 0 < r ≤ n. This formula helps to expand the binomial expressions such as (x + a) 10, (2x + 5) 3, (x - (1/x)) 4, and so on.
www.cuemath.com/algebra/binomial-theorem/Binomial Theorem - Formula, Expansion, Proof, Examples - Cuemath
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What Is n and r in The Binomial Theorem Formula? In the binomial theorem formula of expansion (x+a) n, we use the combinatorics formula that is denoted as n C r, where n is the exponent in the expansion and r is the term number that ranges from 0 to n.
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2024年6月10日 · The binomial theorem is a formula for expanding binomial expressions of the form (x + y) n, where ‘x’ and ‘y’ are real numbers and n is a positive integer. The simplest binomial expression x + y with two unlike terms, ‘x’ and ‘y’, has its exponent 0, which gives a value of 1
In 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 ...
- Exponents
- Exponents of
- The Pattern
- Coefficients
- As A Formula
- Putting It All Together
- Use It
- Geometry
- Isaac Newton
First, a quick summary of Exponents. An exponent says how many timesto use something in a multiplication. An exponent of 1means just to have it appear once, so we get the original value: An exponent of 0means not to use it at all, and we have only 1:
Now on to the binomial. We will use the simple binomial a+b, but it could be any binomial. Let us start with an exponent of 0and build upwards.
In the last result we got: a3 + 3a2b + 3ab2 + b3 Now, notice the exponents of a. They start at 3 and go down: 3, 2, 1, 0: Likewise the exponents of bgo upwards: 0, 1, 2, 3: If we number the terms 0 to n, we get this: Which can be brought together into this: an-kbk How about an example to see how it works:
We are missing the numbers (which are called coefficients). Let's look at all the results we got before, from (a+b)0 up to (a+b)3: And now look at just the coefficients(with a "1" where a coefficient wasn't shown): Armed with this information let us try something new ... an exponent of 4: And that is the correct answer (compare to the top of the pa...
Our next task is to write it all as a formula. We already have the exponents figured out: an-kbk But how do we write a formula for "find the coefficient from Pascal's Triangle"... ? Well, there issuch a formula: It is commonly called "n choose k" because it is how many ways to choose k elements from a set of n. The "!" means "factorial", for exampl...
The last step is to put all the terms together into one formula. But we are adding lots of terms together ... can that be done using one formula? Yes! The handy Sigma Notationallows us to sum up as many terms as we want: Sigma Notation Now it can all go into one formula:
OK ... it won't make much sense without an example. So let's try using it for n = 3: BUT ... it is usually much easier just to remember the patterns: 1. The first term's exponents start at n and go down 2. The second term's exponents start at 0 and go up 3. Coefficients are from Pascal's Triangle, or by calculation using n!k!(n-k)! Like this: We ma...
The Binomial Theorem can be shown using Geometry: In 2 dimensions, (a+b)2 = a2 + 2ab + b2 In 3 dimensions, (a+b)3 = a3 + 3a2b + 3ab2 + b3 In 4 dimensions, (a+b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (Sorry, I am not good at drawing in 4 dimensions!)
As a footnote it is worth mentioning that around 1665 Sir Isaac Newton came up with a "general" version of the formula that is not limited to exponents of 0, 1, 2, .... I hope to write about that one day.
The Binomial Theorem is a quick way (okay, it's a less slow way) of expanding (that is, of multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power. For instance, the expression (3x − 2) is a binomial, 10 is a rather large exponent, and (3x − 2)10 would be very painful to multiply out by hand.
The binomial theorem is an algebraic method of expanding a binomial expression. Essentially, it demonstrates what happens when you multiply a binomial by itself (as many times as you want). For example, consider the expression (4x+y)^7 (4x+ y)7. It would take quite a long time to multiply the binomial (4x+y) (4x+ y) out seven times.
The number of terms in $$\left (a+b\right)^ {n} $$ or in $$\left (a-b\right)^ {n} $$ is always equal to n + 1. Therefore, when n is an even number, then the number of the terms is (n + 1), which is an odd number. When the number of terms is odd, then there is a middle term in the expansion in which the exponents of a and b are the same.
