Binomial Expansion

Front

What is the Binomial Theorem?

Back

The Binomial Theorem provides a formula for the expansion of powers of a binomial, expressed as (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n.

Front

What does (x + 1)^5 expand to using the Binomial Theorem?

Back

x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1.

Front

How do you find the coefficient of a specific term in a binomial expansion?

Back

Use the formula for the binomial coefficient, which is (n choose k) = n! / (k!(n-k)!), where n is the exponent and k is the term number.

Front

What is the general form of the nth term in the expansion of (a + b)^n?

Back

The nth term is given by T(k+1) = (n choose k) * a^(n-k) * b^k.

Front

What is the significance of the coefficients in a binomial expansion?

Back

The coefficients represent the number of ways to choose k elements from n, which corresponds to the number of combinations.

Front

What is the expansion of (x + y)^3?

Back

x^3 + 3x^2y + 3xy^2 + y^3.

Front

How do you determine the fourth term in the expansion of (5 + 3y)^5?

Back

Use the formula T(k+1) = (n choose k) * a^(n-k) * b^k, where n=5, a=5, b=3y, and k=3.

Front

What is the third term in the expansion of (x + 4)^5?

Back

The third term is 160x^3.

Front

What is the fourth term in the expansion of (4x - 3)^5?

Back

The fourth term is -4320x^2.

Front

What is the role of the binomial coefficients in the expansion?

Back

They determine the weight of each term in the expansion based on the number of ways to select terms.