First, in order to learn about polynomials, we need to learn some basic definitions.
Here is the example of a polynomial we were given in class:
Degree of the polynomial: The degree of the polynomial is simply the highest exponent in the equation. So in the example above, the highest exponent is 4, so the degree of the polynomial is 4.
Terms: The terms of the polynomial are each individual piece of the equation that you are adding together. So in the example above, the terms would be , , , -14x and 83.
Standard Form: The standard form of a polynomial is when the polynomial is written with the powers in descending power order. The example is in standard form. It's in standard form because the first term has a power of 4 and then the next has a power of 3 and so on until you get to the last term that has a power of 0.
Coefficients: These are the numbers in front of the variables. In the example, the coefficients are 3,20,-10, -14 and 83.
Leading Coefficient: This is just the number in front of the highest exponent. So in the example the leading coefficient would be 3 because the highest exponent is 4 and 3 is the number in front of it.
Constant: This is the number alone.
Next, we were given names for certain types of polynomial equations. They're classified according to their degree. It would be a good idea to memorize these.
Now, we are going to learn about operations on polynomials and polynomial functions that start on the back side on the notes sheet that you either got in class or you can find on moodle.
First, let and .
1. Evaluate by hand. Give the degree.
a. f(x) + g(x)
In order to solve this, you add the equations together and you simplify. So the answer would be because we added all of the terms together from both equations.
b. f(x) - g(x)
For this one, you just subtract the first equation from the second one which would look like this: and the answer comes out to be
2. Without multiplying the polynomials, predict...
a. the degree of f(x) * g(x)
Make your own prediction of what the answer will be. The class predicted 6.
b. the leading coefficient of f(x) * g(x)
Again make your own prediction of what the leading coefficient will be. The class predicted 10.
3. Now define the function on CAS (menu, 1, 1). Expand f(x) * g(x) and write the result in standard form (menu, 3, 3).
Once you have done these two steps, you should get the answer of
. As you can see here, the prediction for part a of number 2 was wrong. We may have thought that it would be 6 because we were thinking multiplication, but we have to remember what we learned a few chapters ago. We learned that when you are multiplying two terms with exponents, the final answer's exponent will just be the first two exponents added together. So in the example, it is because you just add the 2 in and the 3 in .
4. Generalize to make a rule: The degree of the product of two polynomials is the sum of the degrees of the polynomials. The leading coefficient of the product of two polynomials is the product of the leading coefficients of the polynomials.
5. Continue with f(x) and g(x) in the previous example. Write an unsimplified expression for f(g(x)), predict the degree and predict the leading coefficient.
To write the unsimplified expression, you just plug in g(x) for every x value in f(x) just like we learned in previous chapters. This would be . Then, the degree is going to be 6 because, as we learned in a previous chapter, when you have one exponent raised to another exponent, you multiply them. So since is being squared, the exponent will be 6. Lastly, the leading coefficient is 50 because the number in front of the power of 6 will be 50 once simplified.
6. Expand f(g(x)) on CAS and write it in standard form.
When you do this, you get and we see that we are correct, the degree is 6 and the leading coefficient is 50.
7. Generalize to make another rule: The degree of the composition of 2 polynomials is the product of individual degrees.
This is basically everything that we learned in class as far as taking notes and new material. But, lets try out a few example problems to make sure that you fully understand the new material.
a. list out the coefficients
Think back to what a coefficient is (the number in front of the variables). So based on this, the answer would be 9,24,16.
b. give the degree
When you look back at what a degree is, you will see that it is the highest exponent, so in this case the answer would be 4.
c. give the leading coefficient
Again, look at the definition of a coefficient to see that it is the number in front of the highest exponent. So this number would be 9.
Suppose that f(x) is a polynomial of degree 3, and g(x) is a polynomial of degree 5.
a. What is the degree of f(x) + g(x)?
Between the two polynomials, the highest degree will be so the degree is 5.
b.What is the degree of f(x) * g(x)?
Remember to think back a few chapters ago and think about what we do when we are multiplying terms. When multiplying terms, we must add the exponents together, so 5 + 3 would be 8.
c. What is the degree of f(g(x))?
When raising an exponent to another exponent, don't forget that we have to multiply the two exponents together to get the final power that we are raising to, so 5 * 3 is 15.
That's it! I hope that this explanation make polynomials seem a little less scary and confusing. If this doesn't help, don't forget that you can always go to Mr. Cope for more help.