91Ó°ÊÓ

Pascal's Triangle is a triangular array of numbers that is used to find the coefficients for the terms in a binomial expansion. Each row corresponds to the coefficients of \((a + b)^n\) for different values of \(n\). For instance, in our example, we needed the coefficients for \((a + b)^8\). This corresponds to the 9th row of Pascal's Triangle because we start counting rows from zero.
In Pascal's Triangle:

• The first and last numbers in each row are always 1.
• Any number inside the triangle is the sum of the two numbers directly above it.

Understanding Pascal's Triangle can be very helpful in expanding binomials, and it allows us to find the coefficients without having to manually calculate each binomial coefficient. The 9th row (starting from row 0) in Pascal's Triangle, which we use for \( (a+b)^8 \), is: 1, 8, 28, 56, 70, 56, 28, 8, 1.

Polynomial expansion involves expressing a binomial raised to a power as a sum of terms. Each term in the expansion will be a product of the coefficients (from Pascal's Triangle) and the variable terms raised to appropriate powers. The general form of a polynomial expansion for \( (a + b)^n \) is:

• The terms will have \((n+1)\) terms.
• The exponents of \(a\) start from \((n)\) and decrease to 0.
• The exponents of \(b\) start from 0 and increase to \((n)\).

In our case, expanding \( (a + b)^8 \), we follow these rules:

• The first term has \(a^8\)
• The next term has a's exponent decreasing and b's increasing: \((8a^7b)\), then \((28a^6b^2)\), and so on, until the last term, which is \((b^8)\).

The full expanded form is:
\[a^8 + 8a^7b + 28a^6b^2 + 56a^5b^3 + 70a^4b^4 + 56a^3b^5 + 28a^2b^6 + 8ab^7 + b^8 \]

Coefficients are numbers that multiply the variable terms in polynomial expressions. In the context of binomial expansion, these coefficients are crucial because they dictate the number of times a term is repeated.
When expanding \( (a + b)^8 \) using Pascal's Triangle, the coefficients come from the 9th row, which is: \[\text{1, 8, 28, 56, 70, 56, 28, 8, 1}\]. Each coefficient corresponds to a term in the expanded polynomial:

• The first term has a coefficient of 1, so \(a^8\).
• The second term has a coefficient of 8, so \(8a^7b\).

This process continues until the last term, which also has a coefficient of 1 for \(b^8\). Using these coefficients simplifies finding the expanded form considerably.