Polynomial Basics and TermsDate: 05/20/98 at 20:14:28 From: erin Subject: Algebraic polynomials I'm having a lot of trouble understanding polynomials. My teacher goes through a lesson one chapter a day so it's hard to keep up. If you could, I'd like you to help to explain them to me. Erin Date: 05/22/98 at 00:43:35 From: Doctor Santu Subject: Re: Algebraic polynomials Well, Erin, polynomials are not much more difficult than "x". As you know, in algebra, when we are trying to work backwards to figure out an unknown number from how it turns out and what was done to it, we call the number x, or whatever, and proceed, right? You use polynomials in a similar way. First, you have to know what x squared, and x cubed and things like that are. These are powers of x. Because we have to work with e-mail, I'm going to write x squared, which is just x times x, as x^2 (though when I write on paper I write the 2 small and raised a little bit). Similarly x^3 stands for x*x*x, and so on. Well. polynomials are just combinations of powers of x, like: 7x^5 + x^4 - 4x^3 + 15x^2 + 11 That's basically all there is to what they are. If you want more, here's some vocabulary that goes with polynomials. (If you haven't read or heard any of this, just don't worry. As the great mathematician Tallulah Bankhead said, there's less to this than meets the eye.) The highest power of x in the polynomial is called the degree of the polynomial. The polynomial I wrote above has degree 5. I wrote the polynomial in descending order of powers. Not everyone does that; so in your homework, look carefully to see if there's a high power of x in the middle of the polynomial somewhere. Each separate part of the polynomial that is combined using + and - signs to make up the whole thing is called a term. The above polynomial has 5 terms, starting with 7x^5, sometimes called the leading term because it's in front, then x^4, then -4x^3, and so on. (Minus signs are considered to belong to the term immediately following them.) There are special names for low-powered terms. The term without any x at all, in the above case the 11, is called the constant term. The term with only x is called the linear term (and our example just didn't have one). The term with x^2 is called the quadratic term, (15x^2 for our example). The x^3 term is called the cubic term (-4x^3 in our example), and the fourth-degree term is called the quartic term (x^4 in our example). You can add polynomials, subtract them, and multiply them. Want to see? Here's addition: [3x^4 - 4x^3 - 4x^2 + 7x] + [7x^5 + x^4 - 4x^3 + 15x^2 + 11] = 7x^5 + 4x^4 - 8x^3 + 11x^2 + 7x + 11 What's the secret? The terms in one polynomial basically just ignore all the terms in the other polynomial except for the one with the exact same degree. So the 7x^5, not having any fifth-degree terms in the other polynomial, basically remains the same. The 3x^4 in the first polynomial and the x^4 in the second, get added together to become 4x^4. Think of it as 3 bananas and one banana make 4 bananas. The x^4 behaves as if it were an entirely new unknown as far as addition is concerned. Similarly: -8x^3 comes from adding -4x^3 and -4x^3, 11x^2 comes from adding -4x^2 and 15x^2, 7x is just the 7x from the polynomial on the left, since the polynomial on the right didn't have a term to go with it, and 11 is from the polynomial on the right because the other polynomial didn't have a constant term. Multiplication: (x^2 + 2x + 3) * (4x^2 + 5x + 6) You have to multiply each term from the polynomial on the left by every term on the polynomial on the right, so you get 9 terms in this example. Then you combine them all together to get the answer: 4x^4 + 13x^3 + 28x^2 + 27x + 18 Polynomials can be divided by each other. You can do long division of one polynomial by another, provided the divisor has a degree that's either lower than the one being divided, or maybe equal. Regards, -Doctor Santu, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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