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Questions and Answers
How can the polynomial (x + y)^n be expanded according to the binomial theorem?
How can the polynomial (x + y)^n be expanded according to the binomial theorem?
- As a product of x and y only
- As a single term involving the sum of x and y
- As a sum involving terms of the form ax^by^c (correct)
- As a sum involving terms of the form ax/y
What does the binomial theorem describe in elementary algebra?
What does the binomial theorem describe in elementary algebra?
- The factorization of binomial expressions
- The simplification of binomial fractions
- The multiplication of two binomials
- The algebraic expansion of powers of a binomial (correct)
What are the exponents b and c required to satisfy in the expansion of (x + y)^n?
What are the exponents b and c required to satisfy in the expansion of (x + y)^n?
- b + c = n (correct)
- b - c = n
- b / c = n
- b * c = n
What do the coefficients in the expansion of (x + y)^n form?
What do the coefficients in the expansion of (x + y)^n form?
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In combinatorics, what does (n choose b) represent?
In combinatorics, what does (n choose b) represent?
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Study Notes
Binomial Theorem
- The polynomial (x + y)^n can be expanded according to the binomial theorem, which describes the algebraic expansion of powers of a binomial.
- The binomial theorem is a fundamental concept in elementary algebra that describes the expansion of powers of a binomial into a sum of terms involving various powers of the individual terms.
Exponents b and c
- In the expansion of (x + y)^n, the exponents b and c are required to satisfy the condition b + c = n, where b and c are non-negative integers.
Coefficients in the Expansion
- The coefficients in the expansion of (x + y)^n form a symmetric sequence, with the first and last coefficients being 1 and the coefficients in between being binomial coefficients.
Combinatorics
- In combinatorics, (n choose b) represents the number of ways to choose b items from a set of n items, also known as the binomial coefficient.
- (n choose b) is calculated as n!/b!(n-b)!, where ! denotes the factorial function.
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