Binomial Expansion and Coefficients

The video tutorial explains how to find a single term in a binomial expansion using the binomial theorem. It introduces the formula for finding the r-th term and highlights patterns that simplify the process. Two examples are provided: calculating the fifth term in the expansion of (2x + 3)^7 and the fourth term in the expansion of (3x - 5)^9. The tutorial demonstrates the use of combinations and factorials to simplify calculations, ensuring the sum of exponents equals the binomial's power.

What is the formula used to find a single term in a binomial expansion?

• n choose r times X^(n-r) times Y^r

• n choose r-1 times X^(n-r+1) times Y^(r-1)

• n choose r times X^r times Y^(n-r)

• n choose r-1 times X^r times Y^(n-r+1)

In the binomial expansion, what is the relationship between the exponents of X and Y?

• The sum of the exponents is always n.

• The exponent of X is always greater than Y.

• The exponent of Y is always greater than X.

• The exponents are always equal.

When finding the fifth term in the expansion of (2x + 3)^7, what is the combination used?

• 7 choose 2

• 7 choose 3

• 7 choose 5

• 7 choose 4

What is the value of 7 choose 4 in the calculation of the fifth term?

• 21

• 35

• 42

• 28

In the expansion of (3x - 5)^9, what is the exponent on the negative term for the fourth term?

• 4

• 2

• 3

• 5

What is the combination used to find the fourth term in the expansion of (3x - 5)^9?

• 9 choose 5

• 9 choose 4

• 9 choose 2

• 9 choose 3

What is the value of 9 choose 3 in the calculation of the fourth term?

• 72

• 90

• 56

• 84

What is the result of raising 3x to the power of 6?

• 729x^6

• 243x^6

• 81x^6

• 512x^6

What is the value of negative 5 raised to the power of 3?

• -25

• -125

• -75

• -100

What is the coefficient of the fourth term in the expansion of (3x - 5)^9?

• -5432100

• -7654500

• -6543200

• -8765400