Polynomials: Terms, Exponents, Degrees
Date: 04/18/2002 at 13:37:28 From: Jordan Ransom Subject: Math help like cubic stuff. Can you give me an example of 1. Linear binomial 2. Quadratic trinomial 3. Cubic trinomial 4. Quadratic monomial 5. Six-degree binomial 6. Quadratic binomial 7. Linear monomial 8. Cubic binomial 9. Fourth-degree trinomial 10. Cubic polynomial with four terms. Thanks for the help.
Date: 04/18/2002 at 15:07:23 From: Doctor Ian Subject: Re: Math help like cubic stuff. Hi Jordan, First, let's do a little translation. The following are synonyms: linear <=> first degree quadratic <=> second degree cubic <=> third degree So now the list looks like this: 1. First-degree binomial 2. Second-degree trinomial 3. Third-degree trinomial 4. Second-degree monomial 5. Sixth-degree binomial 6. Second-degree binomial 7. First-degree monomial 8. Third-degree binomial 9. Fourth-degree trinomial 10. Third-degree polynomial with four terms. Now let's do another translation. The following are synonyms: monomial <=> polynomial with 1 term binomial <=> polynomial with 2 terms trinomial <=> polynomial with 3 terms So now the list looks like this: 1. First-degree polynomial with 2 terms 2. Second-degree polynomial with 3 terms 3. Third-degree polynomial with 3 terms 4. Second-degree polynomial with 1 term 5. Sixth-degree polynomial with 2 terms 6. Second-degree polynomial with 2 terms 7. First-degree polynomial with 1 term 8. Third-degree polynomial with 2 terms 9. Fourth-degree polynomial with 3 terms 10. Third-degree polynomial with 4 terms Now it's starting to look kind of like a table. Number of terms Degree 1 2 3 4 First   Second    Third    Fourth  Fifth Sixth  So basically, once you learn how to fill in _any_ slot in the table, you know how to fill _all_ of them in. So even though this looks like a bunch of different kinds of things, they're really all just the same thing, with a couple of knobs (degree and number of terms) that you can tweak. The 'degree' of a term is the sum of the exponents in the term. Here are some examples. Term Exponent(s) Degree ---------- ----------- ------ 2 0 0 [x^0 = 1] 2x 1 1 3x^2 2 2 3x^2y 2,1 3 3xy^2 1,2 3 x^9y^4z^4 9,4,4 17 A polynomial is the sum of a bunch of monomials. (Note that 'poly' means many: a 'polygon' is a shape with many sides, a 'polyglot' is someone who speaks many languages, a 'polytheist' believes in many gods. And 'mono' means 'one': a 'monogamous' person has only one mate, a 'monopoly' is when a product is available from only one vendor, a 'monotheist' believes in only one god.) So here are some polynomials: 2x^2 + 3y x^2y + xy^2 + xz - 2x + 4 x^3 The 'degree' of a polynomial is the _highest_ degree of the monomials that make it up. What can we do with all this? Well, suppose we want to make a fourth degree polynomial with three terms. (This is number 9 on the list.) First, we make spaces for the number of terms we want: ___ + ___ + ___ Then, we make a monomial with the degree of the polynomial: x^4 + ___ + ___ Now just keep adding monomials, being careful not to use an exponent higher than the degree of the polynomial: x^4 + x + 1 So if this has all made sense to you, you can start cranking out the items on your list. I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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