Math Example--Polynomial Concepts--Pascal's Triangle: Example 5

Topic

Polynomials

Description

In this final example, we explore the binomial expansion of (x + y)⁷ using Pascal's Triangle. Our objective is to determine the coefficients for the expanded expression (x + y)⁷ = _x⁷ + _x⁶y + _x⁵y² + _x⁴y³ + _x³y⁴ + _x²y⁵ + _xy⁶ + _y⁷, where the blanks represent the coefficients we need to find.

Pascal's Triangle remains our trusted method for generating coefficients in binomial expansions. Each row of the triangle corresponds to the coefficients of a specific power of a binomial expression. For this example, we utilize the eighth row of Pascal's Triangle (n = 7) to obtain the sequence 1, 7, 21, 35, 35, 21, 7, 1, which provides the coefficients for our expansion.

Examining a series of binomial expansion examples is crucial for students to develop a comprehensive understanding of the concept and its wide-ranging applications. As they work through progressively higher powers and observe the resulting patterns, students enhance their grasp of algebraic structures and combinatorial thinking. These examples highlight the beautiful symmetry and predictability in mathematical patterns, inspiring students to explore more complex polynomial expressions with confidence and enthusiasm.

Teacher's Script: Let's tackle our final example -- the expansion of (x + y)⁷ using Pascal's Triangle. Can you see how the pattern has grown even more? We'll use the row for n = 7, giving us 1, 7, 21, 35, 35, 21, 7, 1. Now, let's apply these coefficients: (x + y)⁷ = _x⁷ + _x⁶y + _x⁵y² + _x⁴y³ + _x³y⁴ + _x²y⁵ + _xy⁶ + _y⁷. Notice how the coefficients are perfectly symmetric. This symmetry is a hallmark of Pascal's Triangle and binomial expansions!

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