Algebra 1 help - Degrees of polynomials

[luvnlattes]

I need some help trying to explain something to my son.

 

Last lesson his book covered degrees of terms in polynomials. Great. He did fine. Today he was supposed put polynomials in descending/ascending order. Because he had just completed the degree lesson, he thought you would use that information in this lesson.

 

For one problem he wrote the descending order as:

-3x⁴y³ + 3x³y³ +5x⁵ - 6x² + 1

 

The answer is supposed to be:

5x⁵ - 3x⁴y³ + 3x³y³ - 6x² + 1

 

He thought you would use the degree of the term to decide how to order them, rather than simply looking for the highest exponent. I explained that is not what you do so he's wondering what is the point of determining the degree of terms in a polynomial?

 

I looked around on the web and everything I find explains to me how to determine the degree but not what it's useful for, so I can't answer his question. Can anyone enlighten us?

 

Thanks so much!

[Dana]

The issue here is really that degree of a polynomial in several variables isn't terribly useful.

In the example you gave, the terms are written in descending order of x.

 

In a polynomial in one variable, the degree tells general shape of the graph and along with the leading coefficient, you can tell end behavior of the graph.

 

In a polynomial in one variable, if you're solving an equation, the degree tells how you'd solve the equation.

If the equation is first degree, the equation is linear and you just isolate the variable.

If the equation is second degree, the equation is quadratic and you solve by setting to zero and factoring or by using the quadratic equation.

A polynomial that's third degree or higher, you solve with other methods.

 

Hope this helps some.

You really only see graphs of polynomials in several variables in calculus when you're graphing.

[luvnlattes]

Okay, so it sounds like this is something that he will use later. It threw him because the two lessons were back to back and the book didn't specify that you were ordering the polynomial according to x. It mentioned it in the example of the teaching part of the lesson,but then went on to other things. The exercises themselves didn't specify the variable to use when ordering.

 

Thank you for your help. :)

[Dana]

Most textbooks do a dreadful job of mentioning when you'll use material later.

I've just been teaching this to my remedial students at the cc.

I find naming polynomials by degree (constant, linear, quadratic, cubic) is far more helpful than by number of terms (monomial, binomial, trinomial). Degree (of polynomial in single variable) gives me a picture of the graph mentally, but terms doesn't.

 

However terms can help with division (divide polynomial by monomial differently than polynomial by something greater than monomial) and with factoring (if binomials follow certain pattern, some special products rules can work, etc).

 

Degree of a polynomial in several variables just doesn't appear much. When factoring, say you had x^2 - 3xy - 4y^2.

Each term has degree 2, but this is the form we'd want to write the polynomial in... descending order of x, ascending order of y. Lets us factor: (x - 4y)(x + y).

 

I don't think I've ever used a text that mentions reasons for naming polynomials or why descending order matters... until you start using leading coefficients for end behavior with graphing in college algebra...

[creekland]

There are times when it is needed to have a polynomial in order - synthetic division comes to mind - but that's in a later course. Knowing the degree of the polynomial tells you how many answers/solutions there are for it. A linear equation has one. A quadratic has two (may be real, repeated at zero, or imaginary). A 5th degree polynomial will have 5 (at least one of which will be real since it's odd - the others can be real, repeated, or imaginary).

 

As stated before, books often do a poor job of letting students know these things ahead of time. I always try to tell my classes even when the usefulness is further down the road. At least then they may have a neuron that remembers hearing it before instead of having it be totally new.