Binomial Theorem Calculator: Expand (ax + b)^n, Find Any Term and Pascal's Triangle
Expand a bracket to a power and find any term.
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How this calculator works
Enter a, b and the power n.
Enter the number of the term you want to find.
The calculator expands the bracket and gives the coefficients, the term you chose and the row of Pascal's triangle.
Formulas used
- (ax + b)^n = \sum_{k=0}^n\binom{n}{k}(ax)^{n-k}b^k
- \binom{n}{k} = \frac{n!}{k!(n - k)!}
- T_{k+1} = \binom{n}{k}(ax)^{n-k}b^k
Worked example
With the default numbers in the calculator above:
- Row of Pascal's triangle (the binomial coefficients, nCk): 1, 5, 10, 10, 5, 1
- The expansion of (ax + b)^n: 32 x^5 + 240 x^4 + 720 x^3 + 1080 x^2 + 810 x + 243
- The coefficients of the powers of x, from the highest power down: 32, 240, 720, 1080, 810, 243
- Term 3 (the general term T(k+1) with k = 2): 720 x^3
- Sum of all the coefficients (put x = 1): 3125
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
What is the binomial theorem?
A formula that expands a bracket of two terms raised to a whole number power as a sum of terms with binomial coefficients.
How do I find a particular term?
Use the general term, which is nCk times the first term to the power n - k times the second term to the power k.
Where does Pascal's triangle come in?
Row n of Pascal's triangle gives the coefficients of the expansion of (x + 1) to the power n.
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Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any school or first-year mathematics text (the binomial theorem).
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