Determining the degree of a polynomial.

Factual Questions
1

I was helping my neighbor’s kid with algebra and it seems his professor has strange ideas about the defintion of the degree of a polynomial.
I always thought the degree of a polynomial was the highest exponent of that polynomial.
For example, the cubic equation
5x[sup]3[/sup] - 2x[sup]2[/sup] + 7x - 10 =0 would be degree 3.
However, his professor says that the degree is the sum of the exponents.
In other words, his professor would classify the above equation as degree 6. Huh ?? :dubious:
The only time I have heard about the summing of exponents to find the degree of an equation would be in the case of 2 or more variables in the polynomial.
For example, the polynomial
y[sup]3[/sup] + x [sup]4[/sup] + 5 y[sup]2[/sup] - 6x = 0 would be degree 7. (summing the highest exponents of the 2 variables).
His professor would say this is degree 10.
Well, what do you math “Dopers” think? Is my definition correct or the professor’s? What surprises me is that the professor has a PhD in mathematics.

(NOTE: Yes, I know the degree of a differential eqation is an entirely different matter so let’s refrain from talking about those).

2

He’s wrong: http://mathworld.wolfram.com/PolynomialDegree.html

3

And what’s this about summing the exponents for the multivariate case? First I’ve ever heard of it.

From here:

4

You’re correct for a polynomial with a single variable–the degree is just the highest power that occurs.

But you’re a bit off for the multi-variable case. In that situation, look at the individual monomials. The degree of a monomial is the sum of the exponents (but you don’t sum degrees from separate monomials).

In your example with x and y, the degree is 4.

Another:

x[sup]6[/sup]y + xy[sup]5[/sup] + x[sup]4[/sup]y[sup]4[/sup]

The degrees of the monomials are 7, 6, and 8, respectively. The degree of the polynomial is the largest of the degrees of the monomials, which is 8.

5

Exactly. I think it’s most likely that the professor said this and the OP’s neighbor’s kid garbled it before telling the OP what was said.

6

Either what the professor said got mixed up or misinterpreted somehow on its way to you, or the professor was smoking crack.

Cabbage is correct. To put it another way, no matter how many variables a polynomial involves, the degree of the polynomial = the degree of the highest-degree term in the polynomial. And the degree of each term is the sum of the powers of the variables in that term. So, for example, the term 7x[sup]2[/sup]yz[sup]5[/sup] has degree 8 (2+1+5). Note that, if the variables were all the same, 7x[sup]2[/sup]xx[sup]5[/sup], this would = 7x[sup]8[/sup], whose degree is obviously 8.

7

I agree with Mathochist re: the most likely source of the confusion.

The sum of the exponents of a polynomial is going to be determined by the highest exponent anyway. Something like x[sup]2[/sup] - 1 should be regarded x[sup]2[/sup] + 0x -1, so every exponent is there every time. You could take the sum of the exponents of the terms with non-zero coefficients, I guess, but it’s not entirely clear that that’s useful.

8

OK, now the more interesting problem becomes, Figure out what the math teacher actually said. Presumably, if he has a PhD in math it was correct, and, like Mathochist says, got garbled in transmission.

I have an idea. Imagine he set his students problems like this:

  1. Find the degree of the following polynomial:

x[sup]2[/sup](x[sup]3[/sup] + 1)

  1. Ditto this one:

y[sup]5[/sup](x[sup]1[/sup] + y[sup]2[/sup])

In each case, since the laws of exponents say that when multiplying you add the exponents, you find yourself adding up the exponents in order to figure out the actual value of the highest power in each case.

(on preview, I see that Thudlow beat me to it, due to my messing with ‘sup’ tags)

9

A nice thing about defining degree this way is that if f and g are polynomials with coefficients coming from an integral domain, then deg(fg)=deg(f)+deg(g). This doesn’t work for the alternate “definition”.

10

Thanks for all the replies so far (and keep these coming if you wish).
I’ll have to verify a test he took in which I am sure the answer to one of the problems involved summing ALL the exponents. I forget the exact problem but
2X[sup]4[/sup] - 9X[sup]3[/sup] + 5X[sup]2[/sup] + 7X -12 =0
is a polynomial of degree 10 according to the professor.
(It might be a few days before I can verify this).
Heck, if the kid doesn’t do well in this course, this would be a good argument that the professor may be lacking in certain math skills himself.

11

and Cabbage (posting#4)
Thanks for your clarification of the multi-variable cases.

12

In my experience, math books are less fallible than math teachers. This is not to say that all math teachers are bad, and I’m sure there are some bad math books out there. But, whatever the result of this little mix-up, perhaps the kid in question will realize that knowing how to read and understand a math textbook is a valuable skill.

13

I’d say it’s likely what Cabbage said: the degree of a monomial is the sum of the exponents that appear. The degree of a polynomial is the highest of the degrees of monomials that appear. Monomial got changed to polynomial in the first sentence and the rest was dropped.