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
Descartes’ 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.)
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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.
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