What if you had two polynomials like and ? How could you add and subtract them? After completing this Concept, you'll be able to perform addition and subtraction on polynomials like these.
Try This
For more practice adding and subtracting polynomials, try playing the Battleship game at http://www.quia.com/ba/28820.html . (The problems get harder as you play; watch out for trick questions!)
Watch This
CK-12 Foundation: 0902S Lesson Addition and Subtraction of Polynomials
Guidance
To add two or more polynomials, write their sum and then simplify by combining like terms.
Example A
Add and simplify the resulting polynomials.
a) Add and
b) Add and and
Solution
a)
b)
To subtract one polynomial from another, add the opposite of each term of the polynomial you are subtracting.
Example B
a) Subtract from
b) Subtract from
Solution
a)
b)
Note: An easy way to check your work after adding or subtracting polynomials is to substitute a convenient value in for the variable, and check that your answer and the problem both give the same value. For example, in part (b) above, if we let and , then we can check as follows:
Since both expressions evaluate to the same number when we substitute in arbitrary values for the variables, we can be reasonably sure that our answer is correct.
Note: When you use this method, do not choose 0 or 1 for checking since these can lead to common problems.
Problem Solving Using Addition or Subtraction of Polynomials
One way we can use polynomials is to find the area of a geometric figure.
Example C
Write a polynomial that represents the area of each figure shown.
a)
b)
c)
d)
Solution
a) This shape is formed by two squares and two rectangles.
To find the total area of the figure we add all the separate areas:
b) This shape is formed by two squares and one rectangle.
To find the total area of the figure we add all the separate areas:
c) To find the area of the green region we find the area of the big square and subtract the area of the little square.
d) To find the area of the figure we can find the area of the big rectangle and add the areas of the pink squares.
To find the total area of the figure we add all the separate areas:
Another way to find this area is to find the area of the big square and subtract the areas of the three yellow squares:
To find the total area of the figure we subtract:
Watch this video for help with the Examples above.
CK-12 Foundation: Addition and Subtraction of Polynomials
Vocabulary
- A polynomial is an expression made with constants, variables, and positive integer exponents of the variables.
- In a polynomial, the number appearing in each term in front of the variables is called the coefficient.
- In a polynomial, the number appearing all by itself without a variable is called the constant.
- Like terms are terms in the polynomial that have the same variable(s) with the same exponents, but they can have different coefficients.
Guided Practice
Subtract from .
Solution:
When subtracting polynomials, we have to remember to subtract each term. If the term is already negative, subtracting a negative term is the same thing as adding:
The final answer is in standard form.
Practice
Add and simplify.
Subtract and simplify.
Find the area of the following figures.