### 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:

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.**

- On the blank screen type the value of
*n*. - Next press the
*math*button on the upper left. - Move the cursor to the right to
*PRB*and select the third option*nCr*. - 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

- Enter the equation Y1 = N
*nCr*X where N is the value of*n*you are looking for. - 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.

THANKYOU

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