Lesson 8 It All Adds Up Develop Understanding

Learning Focus

Add and subtract polynomials algebraically.

Add and subtract polynomials graphically.

How are polynomials like whole numbers?

How is adding and subtracting whole numbers like adding and subtracting polynomials? 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 $132$ . The way we write numbers is really a shortcut because:

$132 = 100 + 30 + 2$

1.

Compare $132$ to the polynomial $x^{2} + 3 x + 2$ . How are they alike? How are they different?

2.

Write a polynomial that is analogous to the number $2, 675$ .

When two numbers are to be added together, many people use a procedure like this:

$\begin{matrix} \begin{matrix} 132 \\ \underset{―}{+ 451} \end{matrix} \\ 583 \end{matrix}$

3.

Write an analogous addition problem for polynomials and find the sum of the two polynomials.

4.

How does adding polynomials compare to adding whole numbers?

5.

Use the polynomials given to find the specified sums in a–f.

$f (x) = x^{3} + 3 x^{2} - 2 x + 10$

$g (x) = 2 x - 1$

$h (x) = 2 x^{2} + 5 x - 12$

$k (x) = - x^{2} - 3 x + 4$

a.

$h (x) + k (x)$

b.

$g (x) + f (x)$

c.

$f (x) + k (x)$

d.

$l (x) + m (x)$

e.

$m (x) + n (x)$

f.

$l (x) + p (x)$

6.

What patterns do you see when polynomials 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:

$\begin{aligned} 368 \\ \underset{―}{- 157} \\ 211 \end{aligned}$

7.

Write the analogous polynomials and subtract them.

8.

Is your answer to #7 analogous to the whole number answer? If not, why not?

9.

Subtracting polynomials 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:

$(x^{3} + x^{2} - 3 x - 5) - (2 x^{3} - x^{2} + 6 x + 8)$

You can line up the polynomials horizontally:

$(x^{3} + x^{2} - 3 x - 5) + (- 2 x^{3} + x^{2} - 6 x - 8)$

$= (- x^{3} + 2 x^{2} - 9 x - 13)$

Or, you can line up the polynomials vertically:

$\begin{matrix} x^{3} + x^{2} - 3 x - 5 \\ \underset{―}{+ (- 2 x^{3} + x^{2} - 6 x - 8)} \\ - x^{3} + 2 x^{2} - 9 x - 13 \end{matrix}$

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 polynomials to find the specified differences in a–d.

$f (x) = x^{3} + 2 x^{2} - 7 x - 8$

$g (x) = - 4 x - 7$

$h (x) = 4 x^{2} - x - 15$

$k (x) = - x^{2} + 7 x + 4$

a.

$h (x) - k (x)$

b.

$f (x) - h (x)$

c.

$f (x) - g (x)$

d.

$k (x) - f (x)$

e.

$l (x) - m (x)$

11.

List three important things to remember when subtracting polynomials.

Ready for More?

Explain why the sum of $(7 x^{2} + 3 x + 2) + (4 x^{2} + 2 x + 1)$ is not analogous to the sum of $732 + 421$ .

Takeaways

Adding Polynomials Algebraically:

Adding Polynomials Graphically:

Subtracting Polynomials:

Vocabulary

polynomial function
subtraction of polynomials
subtrahend
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned to add and subtract polynomials. 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.

Retrieval

1.

Multiply: $4 x (x - 7)$

2.

Multiply $3 x^{2} (x^{2} + 2 x)$