Monday 16 October class

I’m including topics from the Wednesday 11 October class as well, since it never got a proper post!

 

Topics:

Last time: (The link will take you to the web resources and video that I linked for that class)

• Long division using placeholders for missing terms

• Synthetic division

• The relationship between factors of a polynomial and its roots (zeroes): see the three theorems listed in this post.

Notice that you already know the relationship between a factor and a root , from previously using factoring and the zero product property to solve a quadratic equation. For example, to solve , we can factor the left-hand side:

 

 

Use the zero product property:

or 

And we get the roots

or 

Notice that the factor  corresponds to the root 5,

and the factor  corresponds to the root -2.

This is why we write the factor as , so that the root will be .

 

This time:

• Using synthetic division to factor a polynomial (WeBWorK Division of Polynomials problem 9)

• How to check a division by multiplying: we check that

dividend = (quotient)*(divisor) + remainder

We then saw how this leads to the remainder theorem in the case where the divisor has the form x-c: call the dividend (the original polynomial) p(x), and the quotient q(x), then the multiplication check formula says that

p(x) = q(x)*(x-c) + r

If we let x=c in that equation, we get

p(c) = q(c)*(c-c) +r

p(c) = q(c)*0 + r

So p(c) = r

This is the remainder theorem: the remainder you get when you divide p(x) by x-c is just the value p(c) of the polynomial when x=c.

Example: if you would divide by , what would the remainder be? It is not practical to do this division (there are 99 “missing terms” that you would have to replace with 0’s), but we can use the remainder theorem with , and in this case , so the remainder would be

[Note: we also learn from this that is a factor of , since the remainder is 0.]

 

• Connecting factors, roots, and remainders using the three theorems from last time

For a polynomial p(x), the following three things are equivalent (if one of them is true, they all are true):

is a root of

is a factor of 

The remainder is 0 when you divide  by 

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We can check to see if these three things are true by any one of the following means:

• Check to see if  is a root of : is ?

• Check to see if the remainder is 0 when you divide  by  by either long division or synthetic division.

Which of these you will use depends on which is quickest to do and what you will be doing after you answer the question. You should practice all three.

• Graphs of polynomials: basic properties

We examined some examples and learned the following: if p(x) is a polynomial of degree n, then

• p(x) has at most n roots (which are the x-intercepts of its graph)

• The graph of p(x) has at most n-1 “turning points” (Local maxima or minima)

• The end behavior is determined by the leading term of the polynomial:

If the degree is even: both ends go up if the leading coefficient is positive; both ends go down if the leading coefficient is negative. [Think about the graphs of and .]

If the degree is odd: the ends go down to the left and up to the right if the leading coefficient is positive; the reverse if the leading coefficient is negative. [Think about the graphs of  and .]

• For all polynomials, the domain is the set of all real numbers, the graph is continuous everywhere (no jumps, holes, or breaks), and the graph is smooth everywhere (no corners or sharp changes of direction).

These properties will aid us in drawing complete, accurate graphs of polynomials. A complete graph of a polynomial must show all of the features mentioned above, and also show the y-intercept.

 

Homework:

Important! The WeBWorK does not cover everything that you need to know in these topics. Please make sure to do the problems mentioned below!

• Review the examples discussed in class. Make sure that you understand what it is that we are trying to accomplish in each example! In particular, always make sure that you notice whether we are talking about a factor of a polynomial or a root of a polynomial. They are related, but not the same thing.

• It is highly recommended that you learn synthetic division, since we often have to divide by divisors of the form x-c. Please see the link and the video which I linked for last time

(PatrickJMT also has two more example videos for synthetic division which you can find by searching on his website.)

• Do the following problems:

1. If x-7 is a factor of a polynomial p(x), what is the root that corresponds to that factor?
2. If x+1 is a factor of a polynomial p(x), what is the root that corresponds to that factor?
3. If a polynomial p(x) has -64 as a root, what is the factor that corresponds to that root?
4. If a polynomial p(x) has 35 as a root, what is the factor that corresponds to that root?
5. If a polynomial has 0 as a root, what is the factor that corresponds to that root?

Session 8, Exercise 8.2: do it without using either synthetic or long division: rather, use the remainder theorem!

Exercise 8.3: you may use any of the three methods mentioned above to find out if the given g(x) is a factor of f(x): it is highly recommended that you practice all three! And don’t forget to tell the root that goes with the factor.

Exercise 8.4(a-d): I suggest that you use synthetic division to see if the given x-value is a root, since you will want to factor the polynomial if it turns out to be so.

Session 9, Exercise 9.1: be prepared to explain why or why not the graph could be a graph of a polynomial, using correct vocabulary!

Exercises 9.2 and 9.3: match the graphs with their formulas by using the properties of polynomials listed above! (Count the roots & turning points, and look at the end behavior). These exercises are to get you used to thinking about the properties.

• Do the WeBWorK: There are two assignments due tomorrow evening because I’ve extended the one that was due last night. Note: in the assignment about roots and factors and graphs, there are several problems that ask you to match a graph with a polynomial in factored form. Remember that if x=c is a root (x-intercept) of the polynomial, then x-c will be a factor! So look at the x-intercepts and match them with the factors in the polynomials.

• Make sure that your email address is in WeBWorK!

• There will be a quiz next time: the topics will be long division of polynomials and using the remainder theorem.

 

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!