High School Algebra I Unlocked (2016)

Lesson 3.2. Adding and Subtracting Polynomial Expressions

Now that we’ve covered the properties of polynomials, let’s talk about how to add and subtract polynomial expressions.

Whenever you add polynomials together, you want to combine like terms, or terms whose variables and exponents are identical. For example, 4x² and x² are like terms because they have the same variable, x, and the same exponent, 2. On the other hand, 4x² and x are not like terms because even though they have the same variable, they have different exponent values.

The basic process for adding or subtracting polynomials is the following:

1) If necessary, combine polynomials by applying signs through parentheses.

2) Group like terms.

3) Simplify the expression.

Let’s look at an example.

EXAMPLE

Simplify 4x² + x².

Here, just add the terms together:

4x² + x² = 5x²

That’s it: 5x² is the answer.

You can also simplify expressions that have multiple terms of varying degrees. Let’s see how this works in the next question.

EXAMPLE

Express (2x⁴ + x⁶ + 4) + (6x⁶ − x⁴ + 12 − x²) in standard form.

Start by clearing the parentheses in the original expression:

(2x⁴ + x⁶ + 4) + (6x⁶ − x⁴ + 12 − x²) =

2x⁴ + x⁶ + 4 + 6x⁶ − x⁴ + 12 − x²

Next, place like terms next to each other:

2x⁴ + x⁶ + 4 + 6x⁶ − x⁴ + 12 − x² =

x⁶ + 6x⁶ + 2x⁴ − x⁴ − x² + 12 + 4

Finally, combine like terms:

x⁶ + 6x⁶ + 2x⁴ − x⁴ − x² + 12 + 4 =

7x⁶ + x⁴ − x² + 16

That’s all there is to it. When you add together the polynomials (2x⁴ + x⁶ + 4) and (6x⁶ − x⁴ + 12 − x²), the resulting polynomial is expressed in standard form as 7x⁶ + x⁴ − x² + 16.

You perform the same process when you are subtracting polynomials as you do when you are adding polynomials together. Let’s look at how we would subtract polynomials.

Don’t forget to use
PEMDAS to guide the
order in which you
perform the operations.

Parentheses
Exponents
Multiplication
Division
Addition
Subtraction

If you forget the order,
think of the mnemonic
Please Excuse My
Dear Aunt Sally.

EXAMPLE

Simplify (2x⁴ + x⁶ + 4) − (6x⁶ − x⁴ + 12 − x²).

Start by combining the expressions and clearing the parentheses:

(2x⁴ + x⁶ + 4) − (6x⁶ − x⁴ + 12 − x²) =

2x⁴ + x⁶ + 4 − 6x⁶ + x⁴ − 12 + x²

Next, place like terms next to each other:

2x⁴ + x⁶ + 4 − 6x⁶ + x⁴ − 12 + x² =

x⁶ − 6x⁶ + 2x⁴ + x⁴ + x² − 12 + 4

Finally, combine like terms:

x⁶ − 6x⁶ + 2x⁴ + x⁴ + x² − 12 + 4 =

−5x⁶ + 3x⁴ + x² − 8

And you’re done! The result of subtracting the polynomial 6x⁶ − x⁴ + 12 − x² from the polynomial 2x⁴ + x⁶ + 4 is −5x⁶ + 3x⁴ + x² − 8.

You can also add and subtract polynomials that have different variables.

EXAMPLE

Simplify the expression 4f + 5g − 2f² + g² − 2f + 8g in standard form.

First, place like terms next to one another in order of degree:

4f + 5g − 2f² + g² − 2f + 8g =

−2f² + g² + 4f − 2f + 5g + 8g

Then combine like terms to simplify the expression:

−2f² + g² + 4f − 2f + 5g + 8g =

−2f² + g² + 2f + 13g

Therefore, the simplified version of the original expression in standard form is −2f² + g² + 2f + 13g.

That wasn’t too bad, right? Let’s try another one.

EXAMPLE

Simplify the expression (10a³ + b² + a − 10) − (a³ + b² + 2 − a).

There are a lot of terms in this question, but don’t worry. Simply follow the steps for adding and subtracting polynomials, and you’ll do fine.

First, combine the polynomials, applying the negative sign through the second set of parentheses:

(10a³ + b² + a − 10) − (a³ + b² + 2 − a) =

10a³ + b² + a − 10 − a³ − b² − 2 + a

Then place like terms next to one another:

10a³ + b² + a − 10 − a³ − b² − 2 + a =

10a³ − a³ + b² − b² + a + a − 10 − 2

Finally, combine like terms to simplify the expression:

10a³ − a³ + b² − b² + a + a − 10 − 2 =

9a³ + 2a − 12

The simplified version of the original expression is 9a³ + 2a − 12.