Simplifying Polynomials

What Is a Polynomial?

Simplifying a polynomial means rewriting it in its most compact and standard form by combining like terms. But first, what exactly is a polynomial? A polynomial is an algebraic expression made up of one or more terms, where terms are separated by addition or subtraction signs. These terms consist of constants (like $7$), variables (like $x$), or a product of constants and variables raised to non-negative integer powers (like $-4x^2$).

Polynomials are often classified by the number of terms they have:

• Monomial: An expression with one term (e.g., $5x$, $-8$, $2x^{2}y$)
• Binomial: An expression with two terms (e.g., $x+4$, $3y^2-9$)
• Trinomial: An expression with three terms (e.g., $x^2+5x+6$)

To understand polynomials fully, we need to know their basic parts. Let's break down a sample term, $7x^3$:

The degree of a term is its exponent. The degree of a polynomial is the highest degree of any of its terms. For example, in the trinomial $2x^4-5x^2+3$, the degrees of the terms are $4$, $2$, and $0$. The degree of the entire polynomial is $4$.

What Are 'Like Terms'?

The single most important concept for simplifying polynomials is understanding like terms. Think of it like sorting fruit. You can combine apples with other apples and bananas with other bananas, but you can't combine an apple and a banana into a single new fruit. In algebra, like terms are terms that can be combined.

The coefficients of the terms do not have to be the same. Let's look at some examples:

• $5x$ and $-2x$ are like terms. (Variable is $x$, exponent is $1$)
• $8y^2$ and $y^2$ are like terms. (Variable is $y$, exponent is $2$)
• $-3ab$ and $7ab$ are like terms. (Variables are $a$ and $b$, both with exponent $1$)
• $4x^3{y}^2$ and $-x^3{y}^2$ are like terms. (Same variables with same corresponding exponents)

Now, let's see what unlike terms look like:

• $5x$ and $5x^2$ are unlike terms because the exponents of $x$ are different ($1$ vs. $2$).
• $3y$ and $3z$ are unlike terms because the variables are different ($y$ vs. $z$).
• $-2x^{2}y$ and $9x{y}^2$ are unlike terms because the exponents on $x$ and $y$ do not match.

Being able to quickly identify like terms is the key to simplifying. Before you can combine anything, you must first find the terms that match perfectly in their variable parts.

How Do You Combine Like Terms?

Once you have identified the like terms in a polynomial, the process of combining them is straightforward. You simply add or subtract their coefficients and keep the variable part unchanged. It's like saying, "I have 3 apples and I get 4 more apples, so now I have 7 apples." In algebra, that's $3x+4x=7x$. You add the numbers (coefficients) and keep the 'apple' (the variable part).

Let's follow a simple procedure:

1. Scan the polynomial: Look through the entire expression and identify all the groups of like terms. It can be helpful to underline, circle, or use different colors for each group.
2. Group the like terms: Mentally or physically rearrange the expression so that like terms are next to each other. Remember to keep the sign ($+$ or $-$) with its term.
3. Combine the coefficients: For each group of like terms, perform the addition or subtraction on their coefficients.
4. Write the final answer: Write down the new terms to form the simplified polynomial. It's conventional to write the terms in standard form, which means in descending order of their degree.

How Do You Simplify Polynomials with Parentheses?

Often, polynomials are presented with terms grouped inside parentheses. To simplify these, you must first eliminate the parentheses. The tool for this is the distributive property.

This property states that you can multiply the term on the outside of the parentheses by each term on the inside. A very common place for mistakes is when the term on the outside is a negative number or just a subtraction sign. A subtraction sign in front of parentheses is the same as multiplying by $-1$.

For example, $-(2x-3)$ is equivalent to $-1(2x-3)$. Distributing the $-1$ gives $(-1)(2x)+(-1)(-3)=-2x+3$. Notice how the sign of each term inside the parentheses flipped.

After you've used the distributive property to remove all parentheses, the problem becomes a standard one of combining like terms as we did in the previous section.

How Do You Simplify Polynomials with Multiple Variables?

The great news is that nothing changes when you have more than one variable! The rules for identifying and combining like terms are exactly the same. For terms to be 'like', they must have all the same variables, and each of those variables must have the same exponent.

For example, $3x^{2}y$ and $-5x^{2}y$ are like terms because both have $x^2$ and $y^1$. However, $3x^{2}y$ and $3x{y}^2$ are not like terms because the exponents on $x$ and $y$ are different between the two terms.

The key is to be very careful and methodical. Don't rush when checking if the variable parts match. When writing your final answer, standard form is a little less strict, but it's common to order terms alphabetically and by descending degree of the first variable.

A Step-by-Step Simplification Strategy

When faced with a complex polynomial, it's helpful to have a reliable strategy. Follow these steps every time to ensure you get the correct answer without missing anything.

1. Eliminate Parentheses: Your first priority is always to get rid of parentheses. Use the distributive property to multiply the term outside the parentheses by every term inside. Pay close attention to negative signs.
2. Identify and Group: Scan the entire resulting expression. Systematically find all the sets of like terms. Use highlighting, underlining, or circling to keep track of the groups. Write them next to each other.
3. Combine Coefficients: For each group of like terms, add or subtract the coefficients as required. Remember, the variable part of the term does not change during this step.
4. Write the Final Answer in Standard Form: Write out your new, simplified list of terms. For polynomials with a single variable, arrange the terms from the highest exponent to the lowest. For example, an $x^3$ term would come before an $x^2$ term, which comes before an $x$ term, followed by the constant.

Following this ordered process prevents you from getting lost in long expressions and helps you avoid common errors.

Common Mistakes to Avoid

Simplifying polynomials is a process with several places to make small errors. Being aware of these common pitfalls can help you avoid them.

• Combining Unlike Terms: This is the most frequent mistake. Students might accidentally add $5x$ and $3x^2$ to get $8x^3$. Remember, if the variable parts aren't identical, you cannot combine them. They must stay separate.
• Sign Errors with Subtraction: When distributing a negative, it's easy to forget to apply it to all the terms inside the parentheses. For example, in $8x-(2x-5)$, the correct result is $8x-2x+5$, not $8x-2x-5$. The negative must distribute to the $-5$ as well.
• Incorrectly Changing Exponents: When you add or subtract like terms, the exponents do not change. For example, $4x^2+3x^2=7x^2$, NOT $7x^4$. You are simply counting how many $x^2$ terms you have. Exponents only change when you are multiplying or dividing terms, not adding or subtracting.
• Forgetting the Invisible 1: A variable like $x$ or $y^2$ has an invisible coefficient of $1$. So, $5x+x$ is really $5x+1x$, which equals $6x$. Similarly, $-y^2$ is $-1y^2$. Don't let the missing number confuse you.

Quick Summary and Reference

Here's a quick reference guide to the core rules of simplifying polynomials. Use this as a final check when working on problems.

• Goal: To write a polynomial in its most compact form.
• Like Terms: Must have the exact same variables with the exact same exponents.
• The Main Action: Combine like terms by adding or subtracting their coefficients. The variable part stays the same.
• Parentheses Rule: Use the distributive property to eliminate parentheses before you start combining terms.
• Standard Form: Always write your final answer with terms in descending order of degree (e.g., $a{x}^3+b{x}^2+cx+d$).

Mastering these rules will build a strong foundation for all your future work in algebra, from solving equations to graphing functions.