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:(1)
These can be found using the nCr command on your calculator or using Pascal's Triangle.
Why is the Binomial Theorem useful?
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.
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.