Binomial Theorem

Answer

The Binomial Theorem states that for any positive integer n: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + ... + ⁿCₙ₋₁ab^(n-1) + ⁿCₙbⁿ This can also be written as: (a + b)ⁿ = Σ(k=0 to n) ⁿCₖaⁿ⁻ᵏbᵏ…

Answer

Pascal's Triangle is an array of numbers where: • Each row starts and ends with 1 • Each interior number is the sum of the two numbers above it • Row n gives the coefficients for (a + b)ⁿ Example: Ro…

Answer

There are (n + 1) terms in the expansion of (a + b)ⁿ. This means the number of terms is always one more than the power. Examples: • (a + b)² has 3 terms • (a + b)³ has 4 terms • (a + b)⁵ has 6 terms…

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In the expansion of (a + b)ⁿ: • Powers of 'a' decrease by 1 in each successive term: n, n-1, n-2, ..., 1, 0 • Powers of 'b' increase by 1 in each successive term: 0, 1, 2, ..., n-1, n • Sum of powers …

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Binomial coefficients are the numerical coefficients in the binomial expansion, denoted as ⁿCᵣ. Formula: ⁿCᵣ = n!/(r!(n-r)!) These coefficients: • Appear in Pascal's Triangle • Are symmetric: ⁿCᵣ = …

Answer

(x + 2)⁴ = ⁴C₀x⁴ + ⁴C₁x³(2) + ⁴C₂x²(2)² + ⁴C₃x(2)³ + ⁴C₄(2)⁴ = 1·x⁴ + 4·x³·2 + 6·x²·4 + 4·x·8 + 1·16 = x⁴ + 8x³ + 24x² + 32x + 16 Therefore: (x + 2)⁴ = x⁴ + 8x³ + 24x² + 32x + 16…

Answer

(x - y)ⁿ = [x + (-y)]ⁿ = ⁿC₀xⁿ - ⁿC₁xⁿ⁻¹y + ⁿC₂xⁿ⁻²y² - ⁿC₃xⁿ⁻³y³ + ... + (-1)ⁿⁿCₙyⁿ General term: (-1)ʳⁿCᵣxⁿ⁻ʳyʳ The signs alternate: +, -, +, -, ... The pattern depends on (-1)ʳ where r is the te…

Answer

(2x - 3y)³ = (2x)³ - ³C₁(2x)²(3y) + ³C₂(2x)(3y)² - (3y)³ = 8x³ - 3·4x²·3y + 3·2x·9y² - 27y³ = 8x³ - 36x²y + 54xy² - 27y³ Therefore: (2x - 3y)³ = 8x³ - 36x²y + 54xy² - 27y³…