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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 1

Arrange 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 2

Arrange the like terms in columns and add the like terms

Example 1

Let's find the sum of the following two polynomials

(3y5 − 2y + y4 + 2y3 + 5) and (2y5 + 3y3 + 2+ 7)

Subtracting Polynomials

Example 2

Let's find the difference of the same two polynomials

(3y5 − 2y + y4 + 2y3 + 5) and (2y5 + 3y3 + 2+ 7)

Practice Problems

Problem 1

Add the following polynomials: (x3+ 5x + 3x2 +2) and (4x3 + 3x2+ 14)

Problem 2

Find the sum of following polynomials: (2x3+ 5x4 + 3x2 +12) and (7x3 + 4x2+ 3)

Problem 3

Subtract following polynomials: (3x2+ 2x3+ 12x7 + 12) - ( 4x2+ 3 - 11x3 )

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 -11x3 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: (2x8 + 6x7+ 3x9 + 5) + ( 5x2 + 4 + 9x3 )

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 x2) 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.


are all like terms because the variables are all x



are all like terms because the variables are all xy2

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     2x2 + 6x + 5     and     3x2 - 2x - 1

Start with:2x2 + 6x + 5     +     3x2 - 2x - 1
Place like terms together:2x2 + 3x2+6x - 2x+5 - 1
Add the like terms:(2+3)x2+(6-2)x+(5-1)


= 5x2 + 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     (2x2 + 6y + 3xy)  ,   (3x2 - 5xy - x)   and   (6xy + 5)

Line them up in columns and add:

2x2 + 6y + 3xy
3x2      - 5xy - x
           6xy     + 5

5x2 + 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.



PolynomialsIntroduction to AlgebraAlgebra - Basic DefinitionsAlgebra Index

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