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Math Calculators ▶ Binomial Theorem Calculator

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An online binomial theorem calculator helps you to find the expanding binomials for the given binomial equation. No doubt, the binomial expansion calculation is really complicated to express manually, but this handy binomial expansion calculator follows the rules of binomial theorem expansion to provide the best results.

So, come to the point, here you can explore how this calculator expanded the form of the binomial equation and much more!

Read on!

In mathematics, a polynomial that has two terms is known as binomial expression. These two terms will always be separated by either a plus or minus and appears in term of series. This series is known as a binomial theorem. It can also be defined as a binomial theorem formula that arranges for the expansion of a polynomial with two terms.

Furthermore, this theorem is the procedure of extending an expression that has been raised to the infinite power. A binomial theorem calculator can be used for this kind of extension.

It is an algebraic expression that comprises two different terms. For example, \( (a + b), (a^3 + b^3 \), etc.

A binomial expansion calculator automatically follows this systematic formula so it eliminates the need to enter and remember it. The formula is:

- If \( n ∈N,x,y,∈ R \) then

$$^nΣ_{r=0}= ^nC_r x^{n-r} y^r + ^nC_r x^{n-r}· y^r + …………. + ^nC_{n-1}x · y^{n-1}+ ^nC_n · y^n$$

$$ e. (x + y)^n = ^nΣ_r=0 ^nC_rx^{n – r} · yr $$

where,

$$ ^nC_r = n / (n-r)^r $$

it can be written in another way:

$$(a+ b)^n = ^nC_0a^n + ^nC_1a^{n-1}b + ^nC_2a^{n-2}b^2 + ^nC_3a^{n-3}b^3 + … + ^nC_nb^n$$

As indicated by the formula that whenever the power increases the expansion will become lengthy and difficult to calculate. However, a binomial expansion solver can provide assistance to handle lengthy expansions.

Binomial coefficients refer to all those integers that are coefficients in the binomial theorem. Properties of binomial coefficients are given below and one should remember them while going through binomial theorem expansion:

$$ C_0 + C_1 + C_2 + … + C_n = 2n $$

$$ C_0 + C_2 + C_4 + … = C_1 + C_3 + C_5 + … = 2^{n-1} $$

$$ C_0 – C_1 + C_2 – C_3 + … +(−1) ^{n.n}C_n = 0 $$

$$ ^nC_1 + 2. ^nC_2 + 3. ^nC_3 + … + ^{n.n}C_n = n.2^{n-1} $$

$$ C_1 − ^2C_2 + ^3C_3 − ^4C_4 + … +(−1) ^{n-1} C_n = 0 for n > 1 $$

$$ C^0_2 + C^1_2 + C^2_2 + …C^n_2 = (2n)! / (n!)^2 $$

You can use the binomial theorem to expand the binomial. To carry out this process without any hustle there are some important points to remember:

- The number of terms in the expansion of \( (x+y)^n \) will always be \( (n+1) \)
- If we add exponents of x and y then the answer will always be n.
- Binomial coffieicnts are \( ^nC_0, ^nC_1, ^nC_2, … ..,^nC_n \). Anotherr way to represent them is: \( C_0, C_1, C_2, ….., C_n \).
- All those binomial coefficients that are equidistant from the start and from the end will be equivalent. For example: \( ^nC_0 = ^nC_n, ^nC_{1} = ^nC_{n-1} , nC_2 = ^nC_{n-2} ,….. \) etc.

The simplest and error-free way to deal with the expansions is the use of binomial expansion calculator. Furthermore, it also reduces the risk of error that arises from manual calculations.

However, An exponent Calculator can help you to solve the exponent operations and calculate the value of any positive or negative integer raised to the nth power.

Binominal theorem calculator works steadily and quickly. Follow the simple steps explained below:

- first of all, enter a binomial term in the respective filed
- enter the power value
- hit the calculate button

- This binomial series calculator will display your input
- All the possible expanding binomials.

The binomial expansion theorem and its application are assisting in the following fields:

- To solve problems in algebra,
- To prove calculations in calculus,
- It helps in exploring the probability.

However, you can handle the binomial expansion by means of binomial series calculator in all the above-mentioned fields.

This test is commonly used when an experiment has two possible outcomes. It helps you in finding out the probability of success and failure.

- The number of observations that are represented by“n” should be fixed.
- Each observation should be independent.
- Each observation should represent two outcomes i.e. success or failure.
- The probability of “success” p should be the same for every outcome.

Yes! Rolling a die is an example of binomial experiments. It has two outcomes either a number that you have will appear or not. Some other examples are:

- Filliping a coin.
- Asking \( 300 \) people if they watch BBC.

- Mean of the binomial distribution: \( np \)
- Variance of the binomial distribution: \( np (1 − p) \).

If \( p = 0.5 \), then the distribution will be symmetric. But if \( p > 0.5 \), then the distribution will be skewed to the left. On the other hand, if \( p< 0.5 \), then the distribution will be skewed to the right.

This binomial theorem calculator will help you to get a detailed solution to your relevant mathematical problems. You can practice the expansion of binomials to enhance your algebraic skills via this binomial expansion calculator. Therefore, it won’t be wrong to say that it provides assistance to teachers and professors equally and aids them in learning.

- From the source of Wikipedia: Binomial Coefficients, Geometric explanation, combinatorial interpretation.
- From the source of Boundless Algebra: Binomial Expansion and Factorial Notation.
- From the source of Magoosh Math: Binomial Theorem, and Coefficient.