Binomial Theorem - Question 3

WHAT IS BINOMIAL THEOREM?

The expansion of 2 variables to a certain power >0 made simpler with pascals triangle.

Using pascal's triangle to do binomial expansion.

-Each expansion is polynomial.

Some of the patterns observed from the expansion.

1. There is one more term than the power of the exponent, n, when the expression is expanded. there are terms in the expansion of (a + b)ⁿ.

2. In each term, the sum of the exponents is n, the power to which the binomial is raised.

3. The exponents of a start with n, the power of the binomial, and decrease to 0. The last term has no factor of a. The first term has no factor of b, so powers of b start with 0 and increase to n.

4. The coefficients start at 1 and increase through certain values about "half"-way and then decrease through these same values back to 1.

These patterns can be quantified and put into a formula:

IS THERE A POLYNOMIAL THEOREM?

Yes there is.

The polynomial theorem here is referring to the remainder and factor theorem we learnt before.

BINOMIAL THEOREM AND PASCAL'S TRIANGLE.

Compare the below 2 diagrams. Notice any similarities?

Look at the coefficients of the variables in the first diagram. Then, compare them to the numbers of the pascal's triangle. Those that are in the same row have the same coefficients.

Thus, the binomial theorem is related to the pascal's triangle by its coefficients.