Binomial Expansion

How do you find the k-th term in the expansion of (a + b)^n?

• T(k) = (n choose (k-1)) * a^(n-(k-1)) * b^(k-1)

• T(k) = (n choose k) * a^(n-k) * b^k

• T(k) = (n choose (k+1)) * a^(n-k) * b^(k+1)

• T(k) = (n choose (k-2)) * a^(n-(k-2)) * b^(k-2)

How do you determine the third term in the expansion of (x + 4)^5?

• T(3) = (5 choose 2) * x^(5-2) * 4^2 = 160x^3

• T(3) = (5 choose 3) * x^(5-3) * 4^3 = 80x^2

• T(3) = (5 choose 1) * x^(5-1) * 4^1 = 20x^4

• T(3) = (5 choose 0) * x^(5-0) * 4^0 = 1

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

• Σ (n choose k) * a^(n-k) * b^k for k = 0 to n.

• (a + b)^n = a^n + b^n

• (a + b)^n = n * a^(n-1) * b + b^n

• (a + b)^n = a^n + n * a^(n-1) * b + n * a^(n-2) * b^2

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

• It determines the number of terms in the expansion and the highest power of each variable.

• It indicates the coefficients of the terms in the expansion.

• It specifies the order of operations for the expansion.

• It defines the base of the binomial expression.

What is the fourth term in the expansion of (5 + 3y)^5?

• 6750y^3

• 5400y^3

• 8100y^3

• 4050y^3

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

• -4320x^2

• -1080x^2

• 4320x^2

• -540x^2

What is the sum of the coefficients in the expansion of (x + y)^n?

• 2^n

• n^2

• n!

• (x + y)^n

What is Pascal's Triangle and how is it used in binomial expansion?

• A triangular array of prime numbers used in number theory.

• A triangular array of binomial coefficients where each row corresponds to the coefficients in the expansion of (a + b)^n.

• A method for calculating the roots of polynomial equations.

• A geometric representation of the Fibonacci sequence.

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

• x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1

• x^5 + 4x^4 + 6x^3 + 4x^2 + 1

• x^5 + 5x^4 + 15x^3 + 10x^2 + 5x + 2

• x^5 + 3x^4 + 3x^3 + 1

What is the relationship between the terms in the expansion of (x + y)^n?

• The terms are symmetric, meaning T(k) = T(n-k) for any k.

• The terms are independent of each other.

• The terms are always increasing in value.

• The terms can only be positive.