Lesson 2 Adding Your Two Cents Develop Understanding
Learning Focus
Add and subtract linear and quadratic functions algebraically.
Add and subtract linear and quadratic functions graphically.
How are quadratic expressions like whole numbers?
How is adding and subtracting whole numbers like adding and subtracting quadratic expressions? How is it different?
Open Up the Math: Launch, Explore, Discuss
Whenever we’re thinking about algebra and working with variables, it is useful to consider how it relates to the number system and operations on numbers. Let’s see if we can make some useful comparisons between whole numbers and polynomials.
Let’s start by looking at the structure of numbers and polynomials. Consider the number
1.
Compare
2.
Write a polynomial that is analogous to the number
When two numbers are to be added together, many people use a procedure like this:
3.
Write an analogous addition problem for quadratic expressions and find the sum of the two expressions.
4.
How does adding quadratic expressions compare to adding whole numbers?
5.
Use the functions below to find the specified sums in a–f.
a.
b.
c.
d.
e.
f.
6.
What patterns do you see when linear and quadratic functions are added? What patterns do you see when two quadratic functions are added?
Subtraction of whole numbers works similarly to addition. Some people line up subtraction vertically and subtract the bottom number from the top, like this:
7.
Write the analogous quadratic expressions and subtract them.
8.
Is your answer to #7 analogous to the whole number answer? If not, why not?
Subtracting quadratic expressions can easily lead to errors if you don’t carefully keep track of your positive and negative signs. One way that people avoid this problem is to change all the signs of the polynomial being subtracted and then add the two polynomials together. There are two common ways of writing this:
Step 1:
Step 2:
Or, you can line up the expressions vertically so that step 1 looks like this:
9.
Is it correct to change all the signs and add when subtracting? What mathematical property or relationship can justify this action?
10.
Use the given functions to find the specified differences in a–d.
a.
b.
c.
d.
11.
List three important things to remember when subtracting linear and quadratic expressions.
Ready for More?
Explain why the sum of
Takeaways
Adding Linear and Quadratic Polynomials Algebraically:
Adding Linear and Quadratic Polynomials Graphically:
Subtracting Linear and Quadratic Polynomials:
Vocabulary
- polynomial function
- subtrahend
- Bold terms are new in this lesson.
Lesson Summary
In this lesson, we learned to add and subtract polynomials, such as quadratic and linear functions. We learned that the procedure used for adding and subtracting is analogous to adding whole numbers because polynomials have the same structure as whole numbers. Polynomials are added by adding like terms. When subtracting polynomials, we can avoid sign errors by adding the opposite of each term.
Use the Distributive Property to rewrite the expressions.
1.
2.
Label each table as linear, exponential, quadratic, or neither.