Expanding binomials looks complicated, but it’s simply multiplying a binomial by itself a number of times. There is actually a pattern to how the binomial looks when it’s multiplied by itself over and over again, and a couple of different ways to find the answer for a certain exponent or to find a certain part of the resulting polynomial.
Binomials are equations that have two terms. For example, a+b has two terms, one that is “a” and the second that is “b”. Polynomials have more than two terms. Multiplying a binomial by itself will create a polynomial, and the more times it’s multiplied, the longer the resulting polynomial might by. To expand a binomial such as (a+b) to an exponent of 2 is relatively easy. However, when much higher exponents are needed, it’s important to understand the patterns involved and formulas to help get the correct answer.
The Pattern Between Exponents for Expanded Binomials
With a binomial like (a+b), multiplying it by an exponent means multiplying it by itself the number of times specified in the exponent. With an exponent of 2, this becomes a polynomial with 3 parts. With an exponent of 3, it has 4 parts. The higher the exponent, the longer the resulting polynomial.
In the resulting polynomial, a pattern develops with the exponents. For an exponent of 3, the resulting equation has exponents for “a” that start with 3 and go down as the equation is read from left to right. For “b” the exponents start at 0 and go up to 3.
Plugging the numbers into a6(n-k)b(k), where k is the part of the resulting polynomial that’s needed and n is the exponent, it’s possible to determine the exponents of all of the terms of the polynomial. Start with the far left and k=0 to find the exponents for each term of the polynomial.
Determining the Coefficient for Expanded Binomials
Finding the coefficient for these terms involves using Pascal’s Triangle. The first number used is 1, followed by the number of the exponent. From there, following the line in Pascal’s Triangle allows the student to determine the rest of the coefficients in the polynomial.
Another way to find the coefficients is to use the formula n!/(k!(n-k)!). If the binomial being expanded has the exponent of 3 and the 2nd coefficient is the one that is needed, the formula would work as follows: 3!/(2!(3-2)!) = 3!/2!1! = 3x2x1/2x1x1 = 6/2 = 3.
The Binomial Theorem
The binomial theorem is one way to solve for the expansion of a binomial to any exponent. However, it often isn’t going to be the easiest way to solve the problem. If only one term is needed, solving by determining the exponents and solving for the coefficient are going to be easier. However, understanding the Binomial Theorem allows a student to do all of the above calculations at a single time to get the full polynomial from the binomial expansion.
Although working with a binomial with an exponent of 2 or even 3 can be done easily, solving for higher exponents can be more complicated. The formulas here can make it easier for a student to solve for any exponent or find just one term for the resulting polynomial for any exponent.