## High School Algebra I Unlocked (2016)

### Chapter 3. Polynomial Expressions

### 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, 4*x*^{2} and *x*^{2} are like terms because they have the same variable, *x*, and the same exponent, 2. On the other hand, 4*x*^{2} 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 4 x^{2} + x^{2}.**

Here, just add the terms together:

4*x*^{2} + *x*^{2} = 5*x*^{2}

That’s it: 5*x*^{2} 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 (2 x^{4} + x^{6} + 4) + (6x^{6} − x^{4} + 12 − x^{2}) in standard form.**

Start by clearing the parentheses in the original expression:

(2*x*^{4} + *x*^{6} + 4) + (6*x*^{6} − *x*^{4} + 12 − *x*^{2}) =

2*x*^{4} + *x*^{6} + 4 + 6*x*^{6} − *x*^{4} + 12 − *x*^{2}

Next, place like terms next to each other:

2*x*^{4} + *x*^{6} + 4 + 6*x*^{6} − *x*^{4} + 12 − *x*^{2} =

*x*^{6} + 6*x*^{6} + 2*x*^{4} − *x*^{4} − *x*^{2} + 12 + 4

Finally, combine like terms:

*x*^{6} + 6*x*^{6} + 2*x*^{4} − *x*^{4} − *x*^{2} + 12 + 4 =

7*x*^{6} + *x*^{4} − *x*^{2} + 16

That’s all there is to it. When you add together the polynomials (2*x*^{4} + *x*^{6} + 4) and (6*x*^{6} − *x*^{4} + 12 − *x*^{2}), the resulting polynomial is expressed in standard form as 7*x*^{6} + *x*^{4} − *x*^{2} + 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.

**P**arentheses**E**xponents**M**ultiplication**D**ivision**A**ddition**S**ubtraction

If you forget the order,

think of the mnemonic**P**lease **E**xcuse **M**y**D**ear **A**unt **S**ally.

**EXAMPLE **

**Simplify (2 x^{4} + x^{6} + 4) − (6x^{6} − x^{4} + 12 − x^{2}).**

Start by combining the expressions and clearing the parentheses:

(2*x*^{4} + *x*^{6} + 4) − (6*x*^{6} − *x*^{4} + 12 − *x*^{2}) =

2*x*^{4} + *x*^{6} + 4 − 6*x*^{6} + *x*^{4} − 12 + *x*^{2}

Next, place like terms next to each other:

2*x*^{4} + *x*^{6} + 4 − 6*x*^{6} + *x*^{4} − 12 + *x*^{2} =

*x*^{6} − 6*x*^{6} + 2*x*^{4} + *x*^{4} + *x*^{2} − 12 + 4

Finally, combine like terms:

*x*^{6} − 6*x*^{6} + 2*x*^{4} + *x*^{4} + *x*^{2} − 12 + 4 =

−5*x*^{6} + 3*x*^{4} + *x*^{2} − 8

And you’re done! The result of subtracting the polynomial 6*x*^{6} − *x*^{4} + 12 − *x*^{2} from the polynomial 2*x*^{4} + *x*^{6} + 4 is −5*x*^{6} + 3*x*^{4} + *x*^{2} − 8.

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

**EXAMPLE **

**Simplify the expression 4 f + 5g − 2f^{2} + g^{2} − 2f + 8g in standard form.**

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

4*f* + 5*g* − 2*f*^{2} + *g*^{2} − 2*f* + 8*g* =

−2*f*^{2} + *g*^{2} + 4*f* − 2*f* + 5*g* + 8*g*

Then combine like terms to simplify the expression:

−2*f*^{2} + *g*^{2} + 4*f* − 2*f* + 5*g* + 8*g* =

−2*f*^{2} + *g*^{2} + 2*f* + 13*g*

Therefore, the simplified version of the original expression in standard form is −2*f*^{2} + *g*^{2} + 2*f* + 13*g*.

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

**EXAMPLE **

**Simplify the expression (10 a^{3} + b^{2} + a − 10) − (a^{3} + b^{2} + 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:

(10*a*^{3} + *b*^{2} + *a* − 10) − (*a*^{3} + *b*^{2} + 2 − *a*) =

10*a*^{3} + *b*^{2} + *a* − 10 − *a*^{3} − *b*^{2} − 2 + *a*

Then place like terms next to one another:

10*a*^{3} + *b*^{2} + *a* − 10 − *a*^{3} − *b*^{2} − 2 + *a* =

10*a*^{3} − *a*^{3} + *b*^{2} − *b*^{2} + *a* + *a* − 10 − 2

Finally, combine like terms to simplify the expression:

10*a*^{3} − *a*^{3} + *b*^{2} − *b*^{2} + *a* + *a* − 10 − 2 =

9*a*^{3} + 2*a* − 12

The simplified version of the original expression is 9*a*^{3} + 2*a* − 12.