Overview
This lecture explains how to expand binomial expressions using Pascal’s triangle and the binomial theorem, and how to calculate specific term coefficients using combinations.
Expanding Binomials with Pascal’s Triangle
- Binomial expressions like (x - 2)^3 can be expanded using the binomial theorem or by multiplying out.
- Pascal’s triangle provides the coefficients for each term; the row matches the exponent in the binomial.
- For (x - 2)^3, coefficients from the triangle are 1, 3, 3, 1.
- Exponents of x decrease while exponents of -2 increase, summing to the exponent (e.g., 3).
- Expanded terms: x^3, -6x^2, 12x, -8.
Example: Expanding (2x + 3y)^4
- Use the fourth row of Pascal’s triangle: 1, 4, 6, 4, 1.
- Terms are formed as: coefficient × (2x)^(descending power) × (3y)^(ascending power).
- Coefficient calculation incorporates values from binomial components (e.g., 4 × 2^3 × 3).
- Resulting expansion: 16x^4, 96x^3y, 216x^2y^2, 216xy^3, 81y^4.
- Exponents on x decrease; exponents on y increase.
Finding Specific Terms & Coefficients
- To find a specific term (e.g., fourth term), count from the left and apply the binomial formula.
- The coefficient is not just the number from Pascal’s triangle but includes multiplications from the terms.
Combinations and Pascal’s Triangle
- The binomial coefficient nCr gives the value for each position in Pascal’s triangle.
- nCr = n! / [(n - r)! × r!], where n is the exponent and r = term number - 1.
- Calculating nCr shows how to get any coefficient from the triangle, e.g., 6C3 = 20.
General Binomial Term Formula
- The rth term of (a + b)^n: nCr × a^(n - r) × b^r, where r = term number - 1.
- To find a term: identify n (exponent), r (term index minus one), a (first binomial part), b (second part).
Example: (3x - 4y)^6, Fourth and Fifth Term
- Fourth term: use n = 6, r = 3, a = 3x, b = -4y.
- nCr = 6C3 = 20. Term: 20 × (3x)^3 × (-4y)^3 = -34,560 x^3 y^3.
- Fifth term: n = 6, r = 4; 6C4 = 15. Term: 15 × (3x)^2 × (-4y)^4 = 34,560 x^2 y^4.
Key Terms & Definitions
- Binomial Theorem — A formula for expanding expressions of the form (a + b)^n.
- Pascal’s Triangle — A triangular array of binomial coefficients.
- Combination (nCr) — The number of ways to choose r items from n, calculated by n!/(n-r)!r!.
- Coefficient — The numerical part of a term in an algebraic expression.
- Exponent — The power to which a term is raised.
Action Items / Next Steps
- Practice expanding binomials using Pascal’s triangle and the binomial theorem.
- Calculate specific term coefficients using the nCr formula.
- Complete assigned problems on binomial expansions and combinations.