Question

How does the Binomial Theorem’s use Pascal’s triangle to expand binomials raised to positive integer powers?

Answer 1

Answer:

There are  ways for quickly multiply out a binomial that's being raised by an exponent. Like

(a + b)0 = 1

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = (a + b)(a + b)2 = (a + b)(a2 + 2ab + b2) = a3 + 3a2b + 3ab2 + b3

and so on and so on

but there was this mathematician named Blaise Pascal and he found a numerical pattern, called Pascal's Triangle, for quickly expanding a binomial like the ones from earlier. It looks like this

1                           1      

2                   1     2     1

3                1     3     3   1

4           1     4     6     4     1

5       1     5     10     10     5     1

Pascal's Triangle gives us the coefficients for an expanded binomial of the form (a + b)n, where n is the row of the triangle.

Hope this helps!