Math Example--Polynomial Concepts--Pascal's Triangle: Example 4

Topic

Polynomials

Description

This example focuses on the binomial expansion of (x + y)^6 using Pascal's Triangle. We aim to find the coefficients for the expanded expression (x + y)⁶ = _x⁶ + _x⁵y + _x⁴y² + _x³y³ + _x²y⁴ + _xy⁵ + _y⁶, where the blanks represent the coefficients we need to determine.

Pascal's Triangle continues to be an invaluable tool for generating coefficients in binomial expansions. Each row of the triangle corresponds to the coefficients of a specific power of a binomial expression. In this case, we use the seventh row of Pascal's Triangle (n = 6) to obtain the sequence 1, 6, 15, 20, 15, 6, 1, which provides the coefficients for our expansion.

Exploring various examples of binomial expansions is essential for students to fully grasp the concept and its applications. As they work through different powers and observe the resulting patterns, students develop a deeper understanding of algebraic structures and combinatorial principles. These examples showcase the elegant symmetry and predictability in mathematical patterns, encouraging students to approach more complex polynomial expressions with confidence and curiosity.

Teacher's Script: Now, let's examine the expansion of (x + y)⁶ using Pascal's Triangle. Can you see how the pattern is growing? We'll use the row for n = 6, which gives us 1, 6, 15, 20, 15, 6, 1. Let's apply these coefficients: (x + y)⁶ = _x⁶ + _x⁵y + _x⁴y² + _x³y³ + _x²y⁴ + _xy⁵ + _y⁶. Notice how the coefficients are symmetric around the middle term. This is a key feature of Pascal's Triangle that helps us check our work!

For a complete collection of math examples related to Polynomials click on this link: Math Examples: Pascal's Triangle Collection.