Question

How do you expand (4x – 3y)^4# using Pascal’s Triangle?

Answer 1

The Pascal's Triangle is a number triangle where the numbers in each consecutive row going down from the top are the sum of the two numbers directly above it.

Explanation:

The following diagram shows the Pascal's Triangle.

The reason that the triangle relates to the expansion of a binomial is because the coefficients in a given row replace `nC_r`, or `(n, k)` in the binomial expansion formula,

Before expanding your binomial though, we must find the number of terms in the expansion. In a binomial of the form `(a + b)^n`, the number of terms is given by `n + 1`. Therefore there are 5 terms in the expansion.

The numbers in the 5th row of the Pascal's triangle are `1, 4, 6, 4, 1`. We will make the exponents on the 4x descending from 4, and those on the -3y ascending from 0. Don't forget to calculate the exponent on the 4 and the -3. Calculating this we get the following:

`256x^4 - 768x^3y + 864x^2y^2 - 432xy^3 + 81y^4`

Thus, you have your answer. Always make sure to use a test point in your a) original expansion and b) your final answer. I know from experience that errors can very often occur in this process because you will handle big numbers and you will have to calculate a lot. The trick: take your time.

I selected the following test point: `x = 1, y = 1`

In the original expression, this would give `1^4 = 1`.

Calculating in the answer, we get the same thing. Thus, we have the correct answer.

Hopefully this helps!