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.

  1. If you add two even numbers, you’ll always get an even number.
  2. If you subtract an even number from another even number, you’ll always get an even number.
  3. If you add two odd numbers, you’ll always get an odd number.
  4. If you subtract an odd number from another odd number, you’ll always get an odd number.
  5. If you multiply two even numbers, you’ll always get an even number.
  6. If you multiply two odd numbers, you’ll always get an odd number.
  7. If you multiply two integers, you’ll always get an integer.
  8. If you add two integers, you’ll always get an integer.
  9. 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.

  1. If you add two rational numbers, you’ll always get a rational number.
  2. If you multiply two rational numbers, you’ll always get a rational number.
  3. 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.

  1. If you add or subtract two polynomials, will you always get a polynomial?
  2. 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.