The binomial theorem represents , for a nonnegative integer , as , enabling accurate expansion, identification of particular terms or coefficients, and interpretation of as the number of ways to choose which factors contribute . It connects algebraic expansion with combinations and binomial probability; noninteger or negative exponents and infinite-series generalizations are outside this scope.
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Detailed Explanation: Apply the binomial theorem
To expand a power of a binomial, use the binomial theorem:
Here, gives the coefficient, and tells how many factors contribute .
Example: Expand
Identify the parts:
Substitute into the theorem:
Use the binomial coefficients:
Now simplify each term:
Therefore,
The signs alternate here because , so odd powers of are negative.
Learn by doing: Apply the binomial theorem
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Binomial Theorem - Polynomial with Two Integers and Triangle to Expanded Polynomial
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Download/Modify Free Grade 12 'Apply the binomial theorem' Worksheet
Print this free 'Apply the binomial theorem' worksheet to practice this skill away from the computer. Click 'Modify' to easily customize this worksheet to your exact needs.
This Worksheet Includes:
- Binomial Theorem - Polynomial with Two Integers and Triangle to Expanded Polynomial
- Binomial Theorem - Polynomial with Integer and Triangle to Expanded Polynomial
- Binomial Theorem - Polynomial with Variable and Triangle to Expanded Polynomial
- Binomial Theorem - Polynomial with Integer and Power to Binomial Notation and Power
- Binomial Theorem - Polynomial with Integer, Theorem and Power to Binomial Notation and Power
- Binomial Theorem - Polynomial with Integer, Theorem and Power to Value
- Binomial Theorem - Polynomial with Variable and Power to Binomial Notation
- Binomial Theorem - Polynomial with Variable, Theorem and Power to Triangle Value
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