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goldch2 - Contents II Polynomials and Rational Functions...

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Contents II Polynomials and Rational Functions 1 II.A Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II.B Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 II.C Graph of a Rational Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 II.D The Intermediate Value Theorem (IVT) for Rational Functions . . . . . . . . . . . . . . . . 8
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II Polynomials and Rational Functions II.1 II Polynomials and Rational Functions Polynomials are perhaps the most important functions we will deal with. They are easy to manipulate or to evaluate, but finding roots of polynomials is one of the oldest and more difficult tasks in mathematics. We review the definitions and important properties; most of them you have probably seen before. II.A Polynomials A real polynomial is an expression of the form P ( x ) = a n x n + a n - 1 x n - 1 + · · · + a 0 n 0 , n N with a i R and x a variable. The a i are called the coefficients , a 0 is the constant coefficient . If a n 6 = 0 we say P ( x ) has degree n , deg P = n , and a n is called the leading coefficient The zero polynomial P ( x ) = 0, i.e. a n = a n - 1 = · · · = a 0 = 0 poses a bit of a problem: It has no degree, thus no leading coefficient, but constant coefficient zero! To exclude this pesty zero polynomial, we often speak only of “polynomials that have a degree” or of “nonzero polynomials”. If P and Q are nonzero polynomials, then deg ( PQ ) = deg P + deg Q, deg ( P + Q ) = max(deg P, deg Q ) if deg P 6 = deg Q, max(deg P, deg Q ) if deg P = deg Q. If a m is the leading coefficient of P and b n the leading coefficient of Q , then a m b n is the leading coefficient of P · Q . The constant coefficient of P · Q is a 0 b 0 if a 0 is that of P and b 0 that of Q . Examples . 1. The polynomial P ( x ) = 2 x + 1 has degree 1, it is a linear polynomial. Its leading coefficient is 2, its constant coefficient is 1. 2. P ( x ) = x 2 - 5 x + 3 is of degree 2, it is a quadratic polynomial. 3. P ( x ) = πx 3 - x 2 + 2 / 9 is of degree 3, it is a cubic polynomial. 4. P ( x ) = 1124 is of degree 0. It is a constant polynomial. The leading coefficient is 1124 which is also the constant coefficient. 5. If P ( x ) = x 2 + 1, Q ( x ) = - x 2 + 1, then deg P = deg Q = 2, and ( P · Q )( x ) = - x 4 + 1 has degree 4, whereas P ( x ) + Q ( x ) = 2 has degree 0, less than 2 = max(deg P, deg Q ). Polynomials are not always given in expanded form as above. For example, P ( x ) = ( x - 1)( x + 1)( x 2 + 1) is also a polynomial, of degree 4 with leading coefficient 1 and constant coefficient - 1. Its expanded form is P ( x ) = x 4 + 0 · x 3 + 0 · x 2 + 0 · x - 1 = x 4 - 1, as you can check easily. A basic result on polynomials is that one can perform polynomial or long division : Theorem I (Polynomial Division): Let P ( x ) , Q ( x ) be polynomials with deg Q ( x ) = m , (in particular, Q ( x ) is not the zero polynomial!). There are then unique polynomials S ( x ) and R ( x ) such that (i) P ( x ) = S ( x ) Q ( x ) + R ( x )
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II.A Polynomials II.2 and (ii) either R ( x ) = 0 or deg R ( x ) < m . The polynomial R ( x ) is the remainder . The polynomial Q ( x ) is a factor of P ( x ) if and only if R ( x ) = 0.
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