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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:

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

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

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

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.

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

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

## Flashcards covering the Adding and Subtracting Polynomials

Algebra 1 Flashcards