# Lesson 4

Combining Polynomials

- Let's do arithmetic with polynomials.

### 4.1: Notice and Wonder: What Can Happen to Integers

What do you notice? What do you wonder?

- \(7 \boldcdot 9 = 63\)
- \(7 + 9 = 16\)
- \(7 - 9 = \text-2\)
- \(\frac{7}{9} = 0.777 \ldots\)

### 4.2: Experimenting with Integers

Which of these statements are true? Give reasons in support of your answer.

- If you add two even numbers, you’ll always get an even number.
- If you subtract an even number from another even number, you’ll always get an even number.
- If you add two odd numbers, you’ll always get an odd number.
- If you subtract an odd number from another odd number, you’ll always get an odd number.
- If you multiply two even numbers, you’ll always get an even number.
- If you multiply two odd numbers, you’ll always get an odd number.
- If you multiply two integers, you’ll always get an integer.
- If you add two integers, you’ll always get an integer.
- If you subtract one integer from another, you’ll always get an integer.

Which of these statements are true? Give reasons in support of your answer.

- If you add two rational numbers, you’ll always get a rational number.
- If you multiply two rational numbers, you’ll always get a rational number.
- If you divide two rational numbers, you’ll always get a rational number.

### 4.3: Experimenting with Polynomials

Here are some questions about polynomials. You and a partner will work on one of these questions.

- If you add or subtract two polynomials, will you always get a polynomial?
- If you multiply two polynomials, will you always get a polynomial?

- Try combining some polynomials to answer your question. Use the ones given by your teacher or make up your own polynomials. Keep a record of what polynomials you tried, and the results.
- When you think you have an answer to your question, explain your reasoning using equations, graphs, visuals, calculations, words, or in any way that will help others understand your reasons.

### Summary

If we add two integers, subtract one from the other, or multiply them, the result is another integer. The same thing is true for polynomials: combining polynomials by adding, subtracting, or multiplying will always give us another polynomial.

For example, we can multiply \(\text-x^2 + 4.5\) and \(x^3 + 2x + \sqrt7\) to see what happens. We’ll need to use the distributive property, and there are a lot of ways to keep track of the results of distribution when we multiply polynomials. One way is to use a diagram like this:

\(x^3\) | \(2x\) | \(\sqrt{7}\) | |
---|---|---|---|

\(\text-x^2\) | \(\text-x^5\) | \(\text-2x^3\) | \(\text-\sqrt{7}x^2\) |

4.5 | \(4.5x^3\) | \(9x\) | \(4.5\sqrt7\) |

Then we can find the product by adding all the results we filled in. This diagram tells us that the product is \(\text-x^5 + 2.5x^3 - \sqrt{7}x^2 + 9x + 4.5\sqrt7\), which is also a polynomial even though there are square roots as coefficients! No matter what polynomials we started with, multiplying them would give us a polynomial, because we would have to multiply each part of each polynomial and then add them all together. Adding or subtracting polynomials also gives us a polynomial, because we can combine like terms.

When thinking about polynomials, it is important to remember exactly what counts as a polynomial. Any sum of terms that all have the same variable, where the variable is only raised to non-negative integer powers, is a polynomial. So some things that might not look like polynomials at first, like -34.1 or \(7.9998x\), are polynomials.

### Glossary Entries

**degree**The degree of a polynomial in \(x\) is the highest exponent occuring on \(x\) when you write the polynomial out as a sum of non-zero constants times powers of \(x\) (with like terms collected).

**polynomial**A polynomial function of \(x\) is a function given by a sum of terms, each of which is a constant times a whole number power of \(x\). The word polynomial is used to refer both to the function and to the expression defining it.

**relative maximum**A point on the graph of a function that is higher than any of the points around it.

**relative minimum**A point on the graph of a function that is lower than any of the points around it.