Let’s start this lesson with a few things that you’ll have to understand before we start adding and subtracting. A **polynomial** is an expression with more than one term. So what’s a term? A term can be a few things. It can be just a number like:

4 -23 .5 100,000,001

These all qualify as single terms. A term can also be a variable like:

x b g^{2} y^{3}

Lastly, a term can be a collection of the above. This would include:

xy^{2} 45ab^{2}c 9dx^{3} 3.14r^{2}

An addition or a subtraction sign separates terms.

This is one term, or a **monomial**: 6ab^{2}

This would be two terms, or a **binomia**l: 4xy + 8z

This would be three terms, or a **trinomial**: d^{2} + 4d – 3

How many terms does the following have?

-5x^{2} – 14x^{3}y + 4xy^{3} + 11x^{3}

There are four terms!

Ok, got that out of the way! Now, let’s talk about something known as “like terms”. When terms are alike, they can be combined. Terms can be combined when the variables and the exponents of those variables are exactly alike. For example:

** Alike:** 4x

** Not Alike:** -3x

When terms are alike, we can add them together. Here is an example:

(-8x^{4} – 12x^{3}y) + (-5x^{4} – 14x^{3}y + 4xy^{3}) + (-7xy^{3} – 4x^{3}y)

Let’s highlight “like terms” with the same colors.

(-8x^{4} – 12x^{3}y) + (-5x^{4} – 14x^{3}y + 4xy^{3}) + (-7xy^{3} – 4x^{3}y)

We combine these by keeping the variables exactly the same but adding the coefficients (the number in front of the variable) and end up with:

-13x^{4} – 30x^{3}y - 3xy^{3}

**Answer: -13x ^{4} – 30x^{3}y - 3xy^{3}**

And that’s all there is to adding polynomials! When we subtract polynomials, it adds one extra step. When there is a minus sign in front of the parenthesis, we can change it to addition as long as we change every sign in the parenthesis as well. We are technically distributing a “-1” into the parenthesis. For example:

(5x^{3} + 8x^{4}y) – (-7x^{3} – 3x^{2}y^{2} + 12x^{4}y^{4}) + (-7x^{2}y^{2} – 11x^{3})

The second polynomial has a subtraction sign in front, so let’s change that.

(5x^{3} + 8x^{4}y) + (7x^{3} + 3x^{2}y^{2} - 12x^{4}y^{4}) + (-7x^{2}y^{2} – 11x^{3})

Now, let’s locate “like terms”:

(5x^{3} + 8x^{4}y) + (7x^{3} + 3x^{2}y^{2} - 12x^{4}y^{4}) + (-7x^{2}y^{2} – 11x^{3})

If a term doesn’t have any like terms, then it just stays the same value.

x^{3} + 8x^{4}y - 4x^{2}y^{2} - 12x^{4}y^{4}

**Answer: x ^{3} + 8x^{4}y - 4x^{2}y^{2} - 12x^{4}y^{4}**

Below you can **download** some** free** math worksheets and practice.

Simplify each expression.

This free worksheet contains 10 assignments each with 24 questions with answers.**Example of one question:**

**Watch bellow how to solve this example:**

Simplify each expression.

This free worksheet contains 10 assignments each with 24 questions with answers.**Example of one question:**

**Watch bellow how to solve this example:**

Simplify each expression.

This free worksheet contains 10 assignments each with 24 questions with answers.**Example of one question:**

**Watch bellow how to solve this example:**