Binomial Theorem

A Couple Definitions

Binomial - an expression with just two terms. Thus the "bi' part of the nomial. In the case of the binomial theorem we are looking at terms that look like $(a+b)^n$ and the "bi" refers to the a and b before the expansion.

Terms in the Expansion - can be found using the following equation:

\begin{align} \binom{n}{r} a^{n-r}b^{r} \end{align}

These can be found using the nCr command on your calculator or using Pascal's Triangle.

Why is the Binomial Theorem useful?

Example 1

What if n gets too big to reasonably use Pascal's Triangle?

Using a GDC (Ti - 84) to find the values of the coefficients

Steps to calculate the coefficients.

  1. On the blank screen type the value of n.
  2. Next press the math button on the upper left.
  3. Move the cursor to the right to PRB and select the third option nCr.
  4. This will bring you back to the regular screen, type the value of r and press enter… The value should match those in Pascals Triangle. Try a few.

A better way - That I just learned.

This can be done with the table function of the calculator. This allows you to see the entire row of pascals triangle at once! Which means a lot less typing

  1. Enter the equation Y1 = N nCr X where N is the value of n you are looking for.
  2. Press "2nd" and then "Graph" to view the table. You may need to adjust the values displayed in the table. The values of X should be limited to integers.

Example 2

Tougher Example 1

Find the coefficient of the $x^5$ in the expansion of $(3x-2)^8$

Tough Example 2

Find the coefficient of $a^3b^4$ in the expansion of $(5a+b)^7$.

Final Note I have seen IB questions that simply ask you to expand a binomial. But maybe more useful the Binomial theorem can also be useful in finding the derivatives of polynomials from first principles… Check out Derivatives Part 2 It also comes up in discrete probability distributions.

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