Chapter 1: The Binomial Theorem

Question 1

Define binomials and the powers of binomials

Answer 1

Binomial: sum containing 2 terms (a+b)
Power of a binomial: (a+b)ⁿ, which can be expanded through binomial expansion

Question 2

State the 3 rules of factorial notation

Answer 2

1) n! is the product of the first n positive integers
n! = n x (n-1) x (n-2)…x3x2x1
Eg. 5! = 5x4x3x2x1

2) n! = nx(n-1)!
Eg. 8!/6! = 8x7x6!/6! = 8x7 = 56

3) Since 1!=1x0!, 0!=1

Question 3

Explain how to use Pascal’s Triangle to solve binomial expansions

Answer 3

Expansion of (a+b)ⁿ:
- Number of terms of expansion: n+1
- Coefficients (and no. of terms of expansion) are in row n of the Pascal’s Triangle
- Sum of the powers of a and b in each term is n
- From left to right of expansion, powers of a decrease (from aⁿ to a⁰) and powers of b increase (from b⁰ to bⁿ)

Pascal’s Triangle:
n=1: 1,1
n=2: 1,2,1
n=3: 1,3,3,1
n=4: 1,4,6,4,1
n=5: 1,5,10,10,5,1
n=6: 1,6,15,20,15,6,1

Eg. (2+x)³=(2)³+3(2)²(x)+3(2)(x)²+(x)³

Question 4

Explain how the formula for the binomial coefficient and the binomial theorem are used as an alternative to Pascal’s Triangle for binomials of higher powers

Answer 4

Binomial coefficient (nCr):
- The number of possible combinations if there are n objects and r are chosen at 1 time
- The coefficient of aⁿ⁻ʳbʳ in (a+b)ⁿ
nCr = n! / r! (n-r)!

Binomial theorem: finding the general term (Tᵣ₊₁) in (a+b)ⁿ
Tᵣ₊₁= (nCr) aⁿ⁻ʳbʳ

Question 5

Define and find the constant term
Eg. Find the constant term of (x-3/x²)⁹

Answer 5

Constant term: x=0
(refer to notes for answer)

Question 6

Find the coefficient of a term in a binomial expansion
Eg. Find the coefficient of x¹⁰ in (3+2x²)¹⁰

Answer 6

(Refer to notes for answer)