Binomial Theorem

Review: Binomial Expressions

Binomial: Sum of two terms

Examples: (x + y), (2a - 3b), (x + 1)

Powers of binomials: (x + y)², (x + y)³, etc.

Expanding Binomials

Manual expansion (small powers):

(x + y)¹ = x + y

(x + y)² = x² + 2xy + y²

(x + y)³ = x³ + 3x²y + 3xy² + y³

Pattern emerges for larger powers

Binomial Theorem

General formula:

(x + y)ⁿ = Σ C(n,k) xⁿ⁻ᵏ yᵏ for k = 0 to n

Where C(n,k) = n!/(k!(n-k)!) (binomial coefficient)

Expanded form:

(x + y)ⁿ = C(n,0)xⁿ + C(n,1)xⁿ⁻¹y + C(n,2)xⁿ⁻²y² + ... + C(n,n)yⁿ

Example: Expand (x + y)⁴

Using binomial theorem:

(x+y)⁴ = C`(4,0)`x⁴ + C`(4,1)`x³y + C`(4,2)`x²y² + C`(4,3)`xy³ + C`(4,4)`y⁴

Coefficients:
C`(4,0)` = 1
C`(4,1)` = 4
C`(4,2)` = 6
C`(4,3)` = 4
C`(4,4)` = 1

Result:
= x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

Pascal's Triangle

Visual representation of binomial coefficients:

n=0:              1
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

Rule: Each number = sum of two above it

Row n gives coefficients for (x + y)ⁿ

Example: Use Pascal's Triangle

(a + b)⁵ = ?

Row 5: 1, 5, 10, 10, 5, 1

Expand:

= a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

Binomial Coefficients

Properties:

C(n, 0) = C(n, n) = 1 (ends of row)

C(n, k) = C(n, n-k) (symmetry)

C(n, k) + C(n, k+1) = C(n+1, k+1) (Pascal's identity)

Example: Calculate Coefficient

Find coefficient of x³y⁴ in (x + y)⁷

Position: k = 4 (power of y)

Calculate:

C`(7, 4)` = 7!/(4!×3!)
        = 5040/(24×6)
        = 5040/144
        = 35

Coefficient is 35

General Term Formula

kth term (starting from k=0) in (x + y)ⁿ:

Tₖ = C(n, k) xⁿ⁻ᵏ yᵏ

Useful for finding specific term without full expansion

Example: Find Specific Term

Find 4th term in (2x - 3y)⁶

4th term means k = 3 (starting from k=0)

Calculate:

T₃ = C`(6,3)` (2x)⁶⁻³ (-3y)³
   = 20 × (2x)³ × (-27y³)
   = 20 × 8x³ × (-27y³)
   = -4320x³y³

Binomial with Subtraction

(x - y)ⁿ = (x + (-y))ⁿ

Signs alternate: +, -, +, -, ...

Example: Expand (a - b)³

Using binomial theorem:

(a-b)³ = C`(3,0)`a³ + C`(3,1)`a²(-b) + C`(3,2)`a(-b)² + C`(3,3)`(-b)³
       = a³ - 3a²b + 3ab² - b³

Binomial with Coefficients

General form: (ax + by)ⁿ

Apply powers to both coefficient and variable

Example: Expand (2x + 3)⁴

Expand:

= C`(4,0)`(2x)⁴(3)⁰ + C`(4,1)`(2x)³(3)¹ + C`(4,2)`(2x)²(3)² + C`(4,3)`(2x)¹(3)³ + C`(4,4)`(2x)⁰(3)⁴

= 1×16x⁴×1 + 4×8x³×3 + 6×4x²×9 + 4×2x×27 + 1×1×81

= 16x⁴ + 96x³ + 216x² + 216x + 81

Finding Middle Term(s)

If n is even: 1 middle term at position (n/2)

If n is odd: 2 middle terms at positions (n-1)/2 and (n+1)/2

Example: Middle Term

(x + y)⁶ (n=6, even)

Middle position: k = 3

Middle term:

T₃ = C`(6,3)` x³y³
   = 20x³y³

Sum of Coefficients

To find sum of all coefficients:

Set x = 1, y = 1

(1 + 1)ⁿ = 2ⁿ

Example: Sum of Coefficients

In (x + y)⁵:

Coefficients: 1, 5, 10, 10, 5, 1

Sum: 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2⁵ ✓

Alternating Sum of Coefficients

For (x - y)ⁿ, alternating sum:

Set x = 1, y = 1

(1 - 1)ⁿ = 0 (for n > 0)

Confirms alternating sum = 0

Applications: Probability

Binomial distribution: P(k successes) = C(n,k) pᵏ (1-p)ⁿ⁻ᵏ

Example: Coin Flips

Flip fair coin 5 times. Probability of exactly 3 heads?

Calculate:

P(3 heads) = C`(5,3)` (1/2)³ (1/2)²
           = 10 × (1/32)
           = 10/32
           = 5/16

Applications: Approximation

For small x, (1 + x)ⁿ ≈ 1 + nx (first two terms)

Useful in physics and engineering

Example: Approximation

(1.02)⁴ = (1 + 0.02)⁴

First two terms:

≈ 1 + 4(0.02)
= 1 + 0.08
= 1.08

Exact: 1.08243... Close approximation!

Multinomial Theorem

Extension to more than two terms:

(x + y + z)ⁿ = Σ [n!/(i!j!k!)] xⁱ yʲ zᵏ

Where i + j + k = n

Example: Trinomial

(x + y + z)² = x² + y² + z² + 2xy + 2xz + 2yz

Practice