# Adding and Subtracting Polynomials

For the purpose of this article, polynomials are defined as an expression that is made up of variables, constants, coefficients, and exponents that are combined using the mathematical operations of addition and subtraction. Depending on how many terms an expression has, it is called a monomial, binomial, trinomial, or polynomial.

Examples of constants, variables, and exponents are as follows:

- Constants: 1, 2, 5, 38, 67, 104, etc.
- Variables: a, b, c, m, n, x, y, z, etc.
- Coefficients: the 2 in $2x$ , the 5 in $5y$ , etc.
- Exponents: 3 in ${x}^{3}$ , $(n-1)$ in ${13}^{n-1}$ , 2 in ${44}^{2}$ , etc.

## Terms of a polynomial

The terms of the polynomials are the parts of the expression that are generally separated by "+" or "-". So each such part of a polynomial expression is a term.

Example 1

What is the degree of the polynomial $3{x}^{2}+5-4y$ ?

There are three terms: $3{x}^{2}$ , $5$ , and $4y$

The highest exponent on a variable in a polynomial tells us its degree. For example, the polynomial in this example is of degree 2 because the highest power on a variable is 2.

## Adding and subtracting polynomials

If you want to add or subtract polynomials, you have to use the distributive property to add or subtract the coefficients of like terms by factoring. The distributive property states that $x\left(y\right)+x\left(z\right)=x\left(y+z\right)$ .

Like terms are the monomials within a polynomial that have the same variables raised to the same powers, such as $3{x}^{2}y$ and $84{x}^{2}y$ .

Example 2

Add the following polynomials.

$\left(2{x}^{2}+5x+7\right)+\left(3{x}^{2}+2x+5\right)$

First, use the commutative property to group like terms.

$=\left(2{x}^{2}+3{x}^{2}\right)+\left(5x+2x\right)+\left(7+5\right)$

Then use the distributive property and simplify.

$=\left(2+3\right){x}^{2}+\left(5+2\right)x+\left(7+5\right)$

$=5{x}^{2}+7x+12$

Example 3

Subtract the following polynomials.

$\left({x}^{2}+3xy-9\right)-\left(-2{y}^{2}+5xy+6\right)$

You can use the commutative property, but because this is a subtraction problem, you must flip the sign of the terms that are to be subtracted.

$=\left({x}^{2}\right)+\left(3xy-5xy\right)+\left({y}^{2}\right)+\left(-9-6\right)$

Use the distributive property where applicable.

$=\left({x}^{2}\right)+\left(3-5\right)xy+\left(2{y}^{2}\right)+\left(-9-6\right)$

$={x}^{2}-2xy+2{y}^{2}-15$

## Topics related to the Adding and Subtracting Polynomials

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## Get help learning about adding and subtracting polynomials

Learning to add and subtract polynomials can be confusing to many students. If your student could use help figuring out what to do with the constants, variables, and exponents that make up polynomial equations, having them work with a private tutor is an excellent idea. Extra attention outside the classroom can make a world of difference. A tutor can work with your student using their unique learning style so that complex concepts are made easier to understand. To learn more about how tutoring can help your student understand polynomials, contact the Educational Directors at Varsity Tutors today.

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