The Binomial Theorem: Defining Expressions

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  • 0:02 Binomial Theorem
  • 1:21 Why Use Binomial Theorem?
  • 4:38 Binomial Theorem Explained
  • 8:40 Exponents for each Term
  • 10:05 Another Example
  • 12:13 Lesson Summary
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Lesson Transcript
Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

Expert Contributor
Elaine Chan

Dr. Chan has a Ph.D. from the U. of California, Berkeley. She has done research and teaching in mathematics and physical sciences.

In this lesson, students will learn the binomial theorem and get practice using the theorem to expand binomial expressions. The theorem is broken down into its parts and then reconstructed.

Binomial Theorem

Welcome to this lesson on the binomial theorem. While the term sounds complicated, breaking it down into its parts can make it easily understandable:

Since we know that a binomial is a 2-term expression, and a theorem is a mathematical formula, binomial theorem must mean a mathematical formula used to expand 2-term expressions. It is used to find the expanded version of binomials raised to any numerical exponent.

The theorem looks like this:

Binomial theorem
binomial theorem

Before I move on, I'd like to cover three more definitions that will be important:

  • Term is the combination of variables and coefficients separated by operations like + or -
  • Variable is the unknown of a term, usually represented by letters
  • Coefficient is the number being multiplied by the variables

In 3ab + 2b there are two terms, 3ab and 2b; two variables, a and b; and two coefficients, 3 and 2.

Why Use Binomial Theorem?

Back to the formula. Don't let it intimidate you! Even though it seems overly complicated and not worth the effort, the binomial theorem really does simplify the process of expanding binomial exponents. Just think of how complicated it would be to expand 55 manually. You would have to multiply each step on your own:

5 x 5 = 25
5 x 25 = 125
5 x 125 = 625
5 x 625 = 3125

And that is only to the fifth power! Can you imagine how long it takes if you want to go further? I'm certainly glad that calculators have the ability to do this calculation in a fraction of a second. That means I don't have to.

But, if a single number can get so bogged down in the process to expand manually, what if it were a binomial instead? What if we had (a + b)2. We would have to calculate (a + b) (a + b). Well, that isn't so bad; if you are familiar with squaring binomials, you will quickly come to the answer of a2 + 2ab + b2. (If you aren't sure how I got that answer, please review the lessons covering multiplying terms.)

While squaring a binomial isn't terribly difficult or time consuming, what about (a + b)5. Here you would have to do the same thing as we did with the 5: multiply each answer times the original binomial until you have multiplied the binomial by itself five times.

We already have the first two, giving us a result of a2 + 2ab + b2. Now to get (a + b) (a2 + 2ab + b2), we have to remember that we must multiply each term in the binomial by each term in the trinomial (that is the bracket with three terms in it). Let's start with the a term: a * (a2 + 2ab + b2 ) = a3 + 2a2 b + ab2.

Then the b term: b * (a2 + 2ab + b2 ) = a2 b + 2ab2 + b3.

Then we have to combine like terms to get a final result of a3 + 3a2 b + 3ab2 + b3

And that gives us the result for (a + b)3; just two more multiplyings to go, and we are done with raising the binomial to the fifth power! I don't know about you, but I'm already tired of doing this manual process for expanding a binomial exponent, and we've only just finished the third power.

That is what the theorem is good for. It allows us to work out a formula to get straight to our end result instead of having to continually repeat the binomial multiplication.

Binomial Theorem Explained

Let's take a closer look at the formula now.

Binomial theorem
binomial theorem

Starting at the beginning, we have (a + b)n. This just lets us know what power has been chosen for the binomial. n will be defined in each real problem. To define a variable means to assign an actual value to a variable. For our process, let's define n as 3. The equals sign lets us know that the formula is coming next.

Now, that funny angular E is called sigma and means that we will have to repeat a process and sum all the results. Below the sigma, you see k = 0; this just indicates that we will start our summation with the understanding that n can be any number from 0 to whatever it has been defined as. Above the sigma, we see n again. This time, it indicates that the highest exponent is to be used.

Next is what looks like an overgrown parenthesis with a fraction inside. But, the fraction is missing its fraction bar. This is actually the mathematical notation for 'n choose k' and means that you create a fraction out of the factorials of n and k, like this: n! / k!(n - k)! The factorial fraction will give us the coefficient for each term in the final expanded result.

Finally, the remainder of the formula will result in the correct exponents attached to the variables for each term. Notice that the a variable will start at the max value and work its way to zero, while the b variable does the opposite.

Also remember:

  • Anything0 = 1
  • 0! = 1

And that is the formula in a nutshell:

  • Introducing the binomial and its exponent
  • Indicating that multiple results of a process will be summed
  • The coefficient of each term is defined by 'n choose k'
  • The exponents for each term are determined by a pattern of reducing the first by 1 and increasing the second by 1 consecutively

Finding the Coefficient

If we use the example here, of (a + b)3, our coefficient calculations would be:

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Additional Activities

Binomial Theorem Practice Problems

If k is any real number and |x| < 1, then:

(1 + x )k = 1 + kx + k( k - 1) x2 / 2! + k(k - 1)(k - 2) x3 /3! + k(k - 1)(k - 2) (k-3) x4 /4! + ...

Problem 1:

Expand 1 / (1 + x )2 as a power series.

Answer 1:

Use the Binomial Series with k = -2 in the formula given. Since k is a real number, and not a positive integer, the series will be an infinite one. If k had been a positive integer, the series would have been finite. (The formula would have terms that are zero).

1 / (1 + x )2 = 1 - 2x + 3 x2 - 4x3 + 5x4 + ...

Problem 2:

Find the Maclaurin series for the function f(x) = 1 / (4-x)1/2

Answer 2:

The function, f(x), is not yet ready for using the formula given. We need to divide the innermost term by 4 and then multiply by 4 and factor out the 2 to get f(x) ready.

Rewrite the function as 1/2 ( 1 - x/4 )-1/2 .

Use the binomial series with k = -1/2 and x replaced by -x/4.

f(x) = 1/2( 1 + x/8 + 3 x2 / 128 + ... + 1(3)(5)...(2n-1)(x/8)n /n! + ...)

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