How to Expand and Simplify Polynomials?

Polynomials are expressions of variables and constants being connected through basic arithmetic operations. Unlike constants, which only have one expression, a single polynomial has multiple ways of expressing, and you have to choose the best way to express them to help you solve problems related to them. Let’s find out how to expand and simplify polynomials in this article.

Before you learn the specific ways of simplifying and expanding polynomials, you must understand what like terms are. Like terms are terms in a polynomial that have identical variables and powers. However, their coefficients can be different, so the values of like terms are not necessarily equal. If two variables are not like terms, they are called unlike terms.

For instance, 2x and 3x are like terms, while 2x² and 3x³ are not. Alternatively, x and x are like terms, while 7x² and 7y are not.

First of all, we are introducing the process of simplifying polynomials. It’s only a matter of identifying like terms and combining them. For example, if you have a polynomial 5x² + 5x + 9x + 1, you should be able to identify the like terms as 5x and 9x. Then, you can combine them as 14x, such that it becomes 5x² + 14x + 1.

Sometimes, you must adjust the exponent when doing multiplication or division on those terms. In that case, we identify something similar to like terms, but not quite. In this scenario, you can combine polynomials where the variable or the coefficient are the same. For example, if the polynomial is 5x² * 6x + 1, the terms 5x² * 6x can be combined into 30x³, such that the polynomial becomes 30x³ + 1. Or, if the expression you want to simplify is 5 * 7x + 35, you can simplify it as 35x + 35.

But in the example above, we encounter something pretty repetitive. The coefficients of the two terms are exactly the same, so can we simplify it even though it’s not a multiplicative operation? The answer is yes. We use the distributive property, which is expressed by the fact that a(b+c) = ab + ac. Similarly, if you want to simplify 35x + 35 further, you can do the reverse of that operation and simplify it as 35(x+1).

The simplification of polynomials is primarily down to a more convenient front-end display, so even though it is not mathematically necessary, it may be helpful if you want to display that to users.

Expanding polynomials are simply the reverse operation of simplifying it, which we have explained above. However, a few rules are only applicable in the expansion, rather than the simplification, of such expressions. Let’s find out about them in this section.

Firstly, we stick like terms together when we are simplifying polynomials. However, we don’t split the terms while we are expanding them. Attempting to do so brings much ambiguity because there are numerous, if not infinitely many, combinations in which you can add multiple numbers to obtain a single, specific sum. Therefore, we leave the coefficients where they are and should not try to expand them.

When expanding polynomials, you can also see where common identities apply. For example, if you want to expand (x+1)², the result is simply x² + 2x + 1. Similarly, if you encounter the polynomial (x+y)(x-y), you expand that to x² – y². More examples can be found on the identities page on our website. You can also use the distributive law to carry out this task. For example, when you encounter 6(x+y), you can expand it to 6x + 6y.

The expansion of polynomials is commonly used when comparing the actual values of polynomials. For example, these techniques are used to verify whether an equation is an identity or inequality, or to compare the values of two polynomials in mathematical proofs.

In this article, we’ve introduced how we can manipulate polynomials in a helpful way by expanding them and simplifying them. If we missed any critical points that we should have included, please leave them in the comments below so that we can improve this article.