CCSS.MATH.CONTENT.HSA.APR.A.1

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Polynomial Operations in Algebra are key and foundational to the course and for courses like Geometry, Algebra 2, Integrated Maths, Trigonometry, Precalculus, and AP Calculus.

## Polynomial Addition

To add polynomials, all that needs to be done is to combine **like terms**. **Terms** are numbers and/or variables that are separated by a plus or minus sign. Like terms are terms that have the same variable with the same exponent or they’re both constants.

**Constants** are terms that do not have a variable.

Let’s look at an example:

5g^{4} and -g^{4} are like terms. So, these two terms can be combined.

5g^4-g^4=4g^4

Likewise, g^{3 }and 2g^{3}, 2g^{2} and 8g^{2}, and -8g and -5g are like terms.

g^3+2g^3=3g^3

2g^2 +8g^2=10g^2

-8g-5g=-13g

Put all the combined terms together in **standard form** by ordering them from the greatest variable exponent to the least variable exponent.

4g^4+3g^3+10g^2-13g

## Polynomial Subtraction

Subtracting polynomials is pretty similar to adding polynomials. There’s just one beginning step needing to be thrown in there. **Distribution! **Distribute the negative sign to all the terms in the second polynomial. Then, combine the like terms like before!

Check out this example and follow along using the steps as a way to check your progress.

(-7x-6)-(8x^2-2x-1)

Distribute

(-7x-6)+(-8x^2+2x+1)

Combine like terms

-8x^2-5x-5

## Multiplying Polynomials

Multiplying Polynomials uses distribution as well, but this time it’s going to be with terms, not just a negative sign. There are many ways to multiply polynomials. To be honest, there are multiple types of polynomials that will affect how you multiply them.

There are:

**Monomials**: Polynomials with just one term.

**Binomials**: Polynomials with two terms.

**Trinomials**: Polynomials with three terms.

**Etc. Polynomials**: Polynomials with 4 or more terms.

Any type of polynomial can be multiplied by another type of polynomial. The most simple is multiplying two monomials together.

To multiply monomials, multiply the **coefficients** together and multiply the variables together. Coefficients are the numbers directly in front of their respective variables.

(5x)(12x^3)=60x^4

or

-3y^4 (4x^2)=-12x^2y^4

It doesn’t matter how many variables are in a term, they are still considered just one term unless it has a plus or minus sign somewhere in the middle.

To multiply a monomial with a binomial or any other polynomial with any number of terms, just distribute the monomial term to the two terms in the binomial. Multiply the monomial by each term in the binomial.

3x^2(5xy-2x)

15x^3y-6x^3

To multiply a binomial with another binomial, first distribute the first term of the first binomial to the two terms of the second binomial. Then, distribute the second term of the first binomial to the two terms of the second binomial.

(3x+2)(4x-5)

3x(4x-5) = 12x^2 -15x

2(4x-5) = 8x-10

12x^2-15x+8x-10

and simplify

12x^2 -7x -10

Typically, Algebra 1 stops right about here, when multiplying polynomials. When multiplying binomial to trinomials or trinomials to other trinomials, you will use the same concept as with the two binomials multiplied together. Just distribute the first term to the three in the second polynomial. Then the next term. Then the next. Once you’re done with all that multiplying, combine like terms to simplify.

**Closed?**

The standard wants students to know that polynomials are closed under addition, subtraction, and multiplication. All that means is it doesn’t matter how much you multiply, add, or subtract polynomials, by definition, they will remain polynomials.

When dividing polynomials, there’s a chance the result will fail to be a polynomial. Basically, if there’s a fraction in one or more of the polynomial’s terms, it is not a polynomial. Concepts like polynomial division are more seen in Algebra 2 than anywhere else.