MATH 1314, A Functional Approach to College Algebra Frank L. Lewis Chapter 5- Theorems and Rules for Polynomials Section 5.1- Polynomial Function Graphs Intermediate Value Theorem If P(a) and P(b) have opposite signs, then there exists at least one value c between a and b such that P(c)=0. Number of Extrema Rule Let P ( x) = a n x n + a n -1 x n -1 + L + a1 x + a 0 be a polynomial of degree n. Then the graph of P(x) has at most n-1 local extrema (maxima and minima). It can have fewer if some zeros are complex. Section 5.2- Dividing Polynomials Remainder Theorem If P(x) is divided by x-c, then the remainder is the value P(c). Factor Theorem The number c is a zero of P(x) (that is P(c)=0) if and only if x-c is a factor of P(x). Section 5.3- Real Zeros Rational Zeros Theorem If P ( x) = a n x n + a n -1 x n -1 + L + a1 x + a 0 has integer coefficients, then every rational zero is of the form last coefficient factor p factor of a 0 = = q factor of a n first coefficient factor Descartesâ€™ Rule of Signs Let P(x) have real coefficients. 1. The number of positive real zeros is either equal to the number of sign variations in P(x), or less than that by an even whole number. 2. The number of negative real zeros is either equal to the number of sign variations in P(-x), or less than that by an even whole number. Section 5.4- Fundamental Theorem of Algebra Fundamental Theorem of Algebra Every polynomial with complex coefficients has at least one complex zero. (This implies that every polynomial with real coefficients also has at least one complex zero.) 1 Complete Factorization Theorem- Linear Factorization Every polynomial with degree n>0 can be completely factored into n linear factors such that P( x) = a( x - c1 )( x - c 2 ) L ( x - c n ) where the n roots ci are generally complex. Zeros Theorem Every polynomial of degree n>0 has exactly n zeros, provided that a zero of multiplicity k is counted k times. Conjugate Zeros Theorem Let a polynomial have real coefficients. Then if a complex number z is a zero, its complex conjugate z is also a zero. Linear and Quadratic Factors Theorem- Real Factorization Every polynomial with real coefficients and degree n can be factored into a product of linear and quadratic factors, all of which have real coefficients, of the form P ( x) = a ( x - r1 )( x - r2 ) L ( x - rm )( x 2 + b1 x + c1 )( x 2 + b2 x + c 2 ) L where the real zeros are ri and the quadratic factors have complex conjugate zeros. 2

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