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:

(1)

\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?

+ Show Usefulness of Binomial Theorem

The binomial theorem allows the expansion of expressions such as $(x+2)^4$ or $(x+1)^{10}$ with out having to do all of the painful algebra. It allows us to skip many of the tedious algebraic steps where we are also very likely to make mistakes. It is important to remember that the Binomial Theorem is not just some guy behind a curtain doing magic tricks (Wizard of Oz reference). It turns out there is a pattern to expanded binomials. Lets look at the expansion of $(a+b)^n$ for n values from 0 to 4. I have simplified each expression after the expansion.

$1$

$a+b$

$a^2+2ab+b^2$

$a^3 + 3a^2b + 3ab^2+b^3$

$a^4 + 4a^3 b + 6a^2 b^2 + 4a b^3 + b^4$

Notice on each line (but using the last for example) the exponent on the a starts at the value of n (in this case $n=4$) and decrease by one for each term until it the exponent is zero. Notice that the exponent for b does the exact opposite, increasing from 0 to the value of n. This pattern alone is pretty cool in some nerdy sense and remembering the pattern can help you get some points on a tricky IB question.

But there is one more pattern that the "mathies" get particularly excited about. If we look at just the coefficients of the terms they too form a pattern. The pattern is known as Pascal Triangle. The first few lines of his triangle are shown below:

Any number in the triangle can be found by adding the two numbers diagonally above it. For example the 6 is the sum of the two 3's. The 4's are products of the 1 and 3. Knowing this pattern you can quickly continue the pattern. The outside layer of the triangle is always 1's. A nice visualization can be found on wikipedia and copied below

Here's the important bit: Since expansions of binomials follow the same pattern as Pascal's Triangle then we can use that in reverse and Pascal's Triangle allows us to expand binomials. Or in other words less painful algebra to grind through.

Example 1

+ Show Example 1

Expand the expression $(a+b)^6$.

Starting with the 4th row of Pascal's Triangle we can find the 5th and 6th rows to be:

The 6th row gives the coefficients of the terms in the expansion. Remembering how the exponents increase and decrease…

(2)

\begin{equation} a^6 + 6a^5 b+15 a^4b^2+20a^3b^3+15a^2b^4+6ab^5 +b^6 \end{equation}

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

+ Show Big n

Now that's all good and all but what if you want to expand $(a+b)^8$ or $(a+b)^{15}$? Working out Pascal's Triangle that far is doable but time consuming and brings us easily back into the realm of making mistakes. The IB is fine with using the triangle. But you can use your calculator too.

Each term in the expansion of $(a+b)^n$can be written as $\left ( \begin{matrix}n\\ r\end{matrix}\right ) a^{n-r}b^r$. Each term is written by incrementally increasing the value of r from 0 to the value of n.

If $n=4$ then the expansion looks something like:

(3)

\begin{align} \left ( \begin{matrix}4\\ 0\end{matrix}\right ) a^4 + \left ( \begin{matrix}4\\ 1\end{matrix}\right ) a^3b + \left ( \begin{matrix}4\\ 2\end{matrix}\right ) a^2b^2+ \left ( \begin{matrix}4\\ 3\end{matrix}\right ) ab^3+ \left ( \begin{matrix}4\\ 4\end{matrix}\right ) b^4 \end{align}

Everything should look pretty familiar except the coefficient out front of each term. Notice that n is the same value for each term only the value of r has changed. The value of r needs to matches the exponent on b!

So how do you find those coefficients on your calculator?

Using a GDC 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.

Example 2

+ Show Example 2

Use your GDC to find $\binom{3}{2}$

This corresponds to the third term in the fourth row of pascals triangle (note that both n and r start at 0). In the calculator this would look like 3 nCr 2. Which should equal 3.

Tougher Example 1

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

+ Show Answer

In general the term is going to look like:

(4)

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

In this case $a=3x$, $b=-2$ and $n=8$. Since the exponent in the term that we are looking for is 5 then we can state that $n-r=5$. Since $n=8$ then $r=3$.

(5)

\begin{align} \binom{8}{3} (3x)^5 (-2)^3 = \binom{8}{3} (243x^5)(-8) = \binom{8}{3} (-1944) \end{align}

Using a GDC to calculate $\binom{8}{3}=56$. So the coefficient is $(56)(-1944)= -108864$

Tough Example 2

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

+ Show Answer

For this problem $n=7$, $n-r=3$ so $r=4$.

The term will look like:

(6)

\begin{align} \binom{7}{4}(5a)^3b^4 = (35)(125a^3)(b^4)=4375a^3b^4 \end{align}

The answer is just the coefficient which is 4375.

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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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