# Adding and Subtracting Polynomials

The basic component of a polynomial is a **monomial**. When we add or subtract polynomials, we are actually dealing with the addition and subtraction of individual monomials that are similar or alike.

## What is a Monomial?

A monomial can be a single number, a single variable, or the product of a number and one or more variables that contain **whole number exponents**.

This means that the exponents are **neither** negative nor fractional.

**Examples**:

- 7

- y

- 2x

- - \,9xy

- - \,4x{y^2}

- 10{x^2}{y^3}{z^4}

- {3 \over 4}{k^5}{m^2}h{r^{12}}

So now we are ready to define what a polynomial is.

## What is a Polynomial?

A polynomial can be a **single monomial **or a combination of **two or more monomials **connected by the operations of *addition*** **and *subtraction*.

**Examples**:

- 12

- - \,8x

- x + y

- 5{x^2} - 7{y^3}

- 3x - 6y + 2z

- - \,2{m^2} + xyz - 5{k^4}n{p^{10}}

A polynomial has **“special” names** depending on the number of monomials or terms in the expression. More so, the **degree** of a polynomial with a single variable is determined by the largest whole number exponent among the variables.

For example:

### Examples of How to Add and Subtract Polynomials

**Example 1**: Simplify by adding the polynomial expressions

The key in both adding and subtracting polynomials is to make sure that each polynomial is *arranged in standard form*. It means that the **powers of the variables are in decreasing order ***from left to right*.

Observe that each polynomial in this example is already in standard form, so we no longer need to perform that preliminary step.

Now, there are two ways we can proceed from here.

- First, we can add this the “usual” way, that is, add them
**horizontally**.

I suggest that you first **group similar terms in a parenthesis** before performing addition.

- Another way of simplifying this is to add them
**vertically**.

Place the **similar terms in the same column** before performing addition.

As you can see, the answers in both methods came out to be the same!

**Example 2**: Simplify by adding the polynomial expressions

\left( { - \,4{x^4} - 11{x^3} + 6{x^2} - x + 1} \right) + \left( {3x - 6 + 10{x^4} + {x^2} - 7{x^3}} \right)

Notice that the first polynomial is already in the standard form because the exponents are in decreasing order. However, the second polynomial is not! We must first rearrange the powers of x in decreasing order from left to right.

- Then add them horizontally…

Similar or like terms are placed in the same parenthesis.

= \left( { - \,4{x^4} - 11{x^3} + 6{x^2} - x + 1} \right) + \left( {10{x^4} - 7{x^3} + {x^2} + 3x - 6} \right) = \left( { - \,4{x^4} + 10{x^4}} \right) + \left( { - 11{x^3} - 7{x^3}} \right) + \left( {6{x^2} + {x^2}} \right) + \left( { - x + 3x} \right) + \left( {1 - 6} \right) = 6{x^4} - 18{x^3} + 7{x^2} + 2x - 5- Or add them vertically…

Similar or like terms are placed in the same column before performing the addition operation.

**Example 3**: Simplify by adding the polynomial expressions

**Solution:**

We are given two trinomials to add. But first, we have to “fix” each one of them by expressing in standard form.

Add only similar terms. Notice that the y

**Adding Polynomials – Horizontally**

Let’s check our work if the answer comes out the same when we add them vertically.

**Adding Polynomials – Vertically**

**Example 4**: Simplify by adding the polynomial expressions

**Solution:**

Let’s add the polynomials above **vertically**. Align like terms in the same column then proceed with polynomial addition as usual.

**Example 5**: Simplify by subtracting the polynomial expressions

Subtracting polynomials is as easy as changing the operation to normal addition. However, always remember to also switch the signs of the polynomial being subtracted.

This is how it looks when we rewrite the original problem from subtraction to addition with some changes on the signs of each term of the second polynomial.

**Original Problem**

**Rewritten Problem**

- The original subtraction operation is
**replaced by addition**.

- The second polynomial is “tweaked” by reversing the original sign of each term.

At this point, we can proceed with our normal addition of polynomials. Make sure that similar terms are grouped together inside a parenthesis.

**Subtracting Polynomials – Horizontally**

**Subtracting Polynomials – Vertically**

We can also subtract the polynomials in vertical way. First, convert the original subtraction problem into its **addition problem counterpart** as shown by the **green arrow**. Make sure to align similar terms in a column before performing addition.

**Example 6**: Simplify by subtracting the polynomial expressions

\left( { - \,1 + {x^2} - 9{x^4} + 2{x^3} - 3x} \right) - \left( {7{x^2} + 3x - 5 - 6{x^3} + {x^4}} \right)

The two polynomials that we are about to subtract are** not** in standard form. Begin by rearranging the powers of variable x in decreasing order. Change the operation from subtraction to addition, align similar terms and simplify to get the final answer.

Transform each polynomial in standard form

**First Polynomial:**

- { - \,1 + {x^2} - 9{x^4} + 2{x^3} - 3x} (not in standard form)

- - \,9{x^4} + 2{x^3} + {x^2} - 3x - 1 (standard form)

**Second Polynomial:**

- {7{x^2} + 3x - 5 - 6{x^3} + {x^4}} (not in standard form)

- {x^4} - 6{x^3} + 7{x^2} + 3x - 5 (standard form)

Subtract by switching the signs of the second polynomial, and then add them together.

**Example 7**: Simplify by subtracting the polynomial expressions

In this problem, we are going to perform the subtraction operation twice.

That means we also need to flip the signs of the two polynomials which are the second and third.

Perform regular addition using columns of similar or like terms.

**Example 8**: Simplify by adding and subtracting the polynomials

\left( {7{y^3} - 10{y^2} + y - 3} \right) - \left( {5 - 4y + {y^2}} \right) + \left( { - \,7{y^3} + 3y} \right)

**Solution:**

Rewrite each polynomial in standard format. Replace subtraction by addition while reversing the signs of the polynomial in question. Finally, organize like or similar terms in the same column and proceed with regular addition.

Adding the polynomials vertically, we have…

**Example 9**: Simplify by adding and subtracting the polynomials

\left( { - x + 6{x^3} - 5} \right) + \left( {7 - {x^2} - 4x} \right) + \left( { - \,{x^3} - 6x + 13} \right)

**Solution:**

Rewrite each polynomial in standard format. Replace subtraction by addition while reversing the signs of the polynomial in question. Finally, organize like or similar terms in the same column and proceed with regular addition.

If we add the polynomials vertically, we have…

**You might also be interested in:**

Dividing Polynomials using Long Division Method

Dividing Polynomials using Synthetic Division Method

Multiplying Binomials using FOIL Method

Multiplying Polynomials