Adding and subtracting

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                                  g2                                y3


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


xy2                              45ab2c                                   9dx3                           3.14r2



An addition or a subtraction sign separates terms.

This is one term, or a monomial:                                   6ab2

This would be two terms, or a binomial:                       4xy + 8z

This would be three terms, or a trinomial:                    d2 + 4d – 3


How many terms does the following have?

-5x2 – 14x3y + 4xy3 + 11x3

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:                       4x2y and -3x2y                      5a2b3c and 2a2b3c                    -4 and 6

Not Alike:                -3x2y and 2xy                       3a2b3c and 5a3b2c                   6yz and 6z


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

(-8x4 – 12x3y) + (-5x4 – 14x3y + 4xy3) + (-7xy3 – 4x3y)

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


(-8x4 – 12x3y) + (-5x4 – 14x3y + 4xy3) + (-7xy3 – 4x3y)


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:


-13x4 – 30x3y - 3xy3

Answer: -13x4 – 30x3y - 3xy3


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:


(5x3 + 8x4y) – (-7x3 – 3x2y2 + 12x4y4) + (-7x2y2 – 11x3)


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


(5x3 + 8x4y) + (7x3 + 3x2y2 - 12x4y4) + (-7x2y2 – 11x3)


Now, let’s locate “like terms”:


(5x3 + 8x4y) + (7x3 + 3x2y2 - 12x4y4) + (-7x2y2 – 11x3)


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


x3 + 8x4y - 4x2y2 - 12x4y4


Answer: x3 + 8x4y - 4x2y2 - 12x4y4

Below you can download some free math worksheets and practice.

1629 x

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:


1420 x

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:


1335 x

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:


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Algebra and Pre-Algebra

Beginning Algebra
Adding and subtracting integer numbers
Dividing integer numbers
Multiplying integer numbers
Sets of numbers
Order of operations
The Distributive Property
Verbal expressions
Beginning Trigonometry
Finding angles
Finding missing sides of triangles
Finding sine, cosine, tangent
Absolute value equations
Distance, rate, time word problems
Mixture word problems
Work word problems
One step equations
Multi step equations
Graphing exponential functions
Operations and scientific notation
Properties of exponents
Writing scientific notation
By grouping
Common factor only
Special cases
Linear Equations and Inequalities
Plotting points
Graphing absolute value equations
Percent of change
Markup, discount, and tax
Adding and subtracting
Quadratic Functions
Completing the square by finding the constant
Solving equations by completing the square
Solving equations by factoring
Solving equations by taking square roots
Solving equations with The Quadratic Formula
Understanding the discriminant
Absolute value inequalities
Graphing Single Variable Inequalities
Radical Expressions
Adding and subtracting
Simplifying single radicals
The Distance Formula
The Midpoint Formula
Rational Expressions
Adding and subtracting
Multiplying and dividing
Simplifying and excluded values
Systems of Equations and Inequalities
Graphing systems of inequalities
Solving by elimination
Solving by graphing
Solving by substitution
Word problems