8.4 The Binomial THeorem
A binomial is a polynomial with two terms.(x + y)ⁿ
for several values of n.
(x + y)⁰ = 1
(x + y)¹ = x + y
(x + y)² = x² +2xy + y²
(x + y)³ = x³ + 3x²y + 3xy² + y³
(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ +y⁴
(x + y)⁵ = x⁵ + 5x⁴y + 10x³y² +10x²y³ + 5xy⁴ + y⁵
(x + y)⁶ = x⁶ + 6x⁵y + 15x⁴y² + 20x³y³ + 15x²y⁴ + 6xy⁵ + y⁶
There are several observations that can be made:
The coefficients in a binomial expansion are called binomial coefficients. To find them, the binomial theorem can be employed.
for several values of n.
(x + y)⁰ = 1
(x + y)¹ = x + y
(x + y)² = x² +2xy + y²
(x + y)³ = x³ + 3x²y + 3xy² + y³
(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ +y⁴
(x + y)⁵ = x⁵ + 5x⁴y + 10x³y² +10x²y³ + 5xy⁴ + y⁵
(x + y)⁶ = x⁶ + 6x⁵y + 15x⁴y² + 20x³y³ + 15x²y⁴ + 6xy⁵ + y⁶
There are several observations that can be made:
- There are n + 1 terms in each expansion.
- In each expansion, x and y have symmetric roles. The powers of x decrease by 1 in successive terms, whereas the powers of y increase by 1.
- The sum of the powers in each term is n.
- The coefficients increase and decrease in a symmetric pattern.
The coefficients in a binomial expansion are called binomial coefficients. To find them, the binomial theorem can be employed.
The Binomial Theorem
In the expansion of (x + y)ⁿ
(x + y)ⁿ = xⁿ + nxⁿ ⁻ ¹y + ⋅⋅⋅ + nCrxⁿ⁻r yⁿ⁻r
the coefficient of xⁿ⁻r yr is
nCr = n!/((n-r)!r!
In the expansion of (x + y)ⁿ
(x + y)ⁿ = xⁿ + nxⁿ ⁻ ¹y + ⋅⋅⋅ + nCrxⁿ⁻r yⁿ⁻r
the coefficient of xⁿ⁻r yr is
nCr = n!/((n-r)!r!
Finding Binomial Coefficients
It is a rule that nCr = nCn - r
There is an easy way to expand binomials.
There is an easy way to expand binomials.
Expanding a Binomial
Sometimes it is needed to find a specific term in a binomial expansion. Instead of writing out the entire expansion, the fact from the Binomial Theorem, the (r + 1)th therm is nCrxⁿ⁻r yⁿ⁻r
Finding a Term or Coefficient in a Binomial Expansion
Pascal's Triangle