## Adding And Subtracting Polynomials Homework Help

Whether you want to add polynomials or subtract them, you follow a similar set of steps.

#### Adding Polynomials

Step 1Arrange the Polynomial in standard form

Standard form of a polynomial just means that the term with highest degree is first and each of the following terms

Step 2Arrange the like terms in columns and add the like terms

##### Example 1

Let's find the sum of the following two polynomials

(3y^{5}− 2y + y

^{4}+ 2y

^{3}+ 5) and (2y

^{5}+ 3y

^{3}+ 2+ 7)

#### Subtracting Polynomials

##### Example 2

Let's find the difference of the same two polynomials

(3y^{5}− 2y + y

^{4}+ 2y

^{3}+ 5) and (2y

^{5}+ 3y

^{3}+ 2+ 7)

**Practice** Problems

##### Problem 1

Add the following polynomials: (x^{3}+ 5x + 3x^{2} +2) and (4x^{3} + 3x^{2}+ 14)

##### Problem 2

Find the sum of following polynomials: (2x^{3}+ 5x^{4} + 3x^{2} +12) and (7x^{3} + 4x^{2}+ 3)

##### Problem 3

Subtract following polynomials: (3x^{2}+ 2x^{3}+ 12x^{7} + 12) - ( 4x^{2}+ 3 - 11x^{3} )

This problem is like example 2 since we are subtracting.

First, remember to rewrite each polynomial in standard form, line up the columns and add the like terms.

(Be careful with -11x^{3}term, since it is already negative, when you subtract the term becomes positive as you can see in the work below.)

##### Problem 4

Add following polynomials: (2x^{8} + 6x^{7}+ 3x^{9} + 5) + ( 5x^{2} + 4 + 9x^{3} )

Although this problem involves addition, there are no like terms. If you line up the polynomials in columns, you will see that no terms are in the same columns.

## Adding and Subtracting Polynomials

A polynomial looks like this:

example of a polynomial this one has 3 terms |

To add polynomials we simply add any **like terms** together .. so what is a like term?

## Like Terms

Like Terms are **terms** whose variables (and their exponents such as the 2 in x^{2}) are the same.

In other words, terms that are "like" each other.

Note: the **coefficients** (the numbers you multiply by, such as "5" in 5x) can be different.

### Example:

are all **like terms** because the variables are all **x**

### Example:

(1/3)xy^{2} | -2xy^{2} | 6xy^{2} | xy/2^{2} |

are all **like terms** because the variables are all **xy ^{2}**

### Example: These are **NOT** like terms because the variables and/or their exponents are different:

## Adding Polynomials

Two Steps:

- Place
**like terms**together - Add the like terms

Example: Add **2x ^{2} + 6x + 5** and

**3x**

^{2}- 2x - 1Start with: | 2x + ^{2} + 6x + 53x^{2} - 2x - 1 | ||||

Place like terms together: | 2x^{2} + 3x^{2} | + | 6x - 2x | + | 5 - 1 |

Add the like terms: | (2+3)x^{2} | + | (6-2)x | + | (5-1) |

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

Here is an animated example:

*(Note: there was no "like term" for the -7 in the other polynomial, so we didn't have to add anything to it.*)

## Adding in Columns

We can also add them in columns like this:

## Adding Several Polynomials

We can add several polynomials together like that.

Example: Add **(2x ^{2} + 6y + 3xy)** ,

**(3x**and

^{2}- 5xy - x)**(6xy + 5)**

Line them up in columns and add:

2x^{2} + 6y + 3xy

3x^{2} - 5xy - x__ 6xy + 5__

5x^{2} + 6y + 4xy - x + 5

Using columns helps us to match the correct terms together in a complicated sum.

## Subtracting Polynomials

To subtract Polynomials, first **reverse the sign of each term** we are subtracting (in other words turn "+" into "-", and "-" into "+"), **then add** as usual.

Like this:

*Note: After subtracting 2xy from 2xy we ended up with 0, so there is no need to mention the "xy" term any more.*

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