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Theorems on the Roots of Polynomial Equations Division Algorithm:

If p(x) and d(x) are any two nonconstant polynomials then there are unique polynomials q(x) and R(x) such that p(x) = d(x) q(x) + R(x) where R(x) is either zero or it is of lower degree than d(x).

Remainder Theorem:

If the polynomial p(x) is divided by (x – r) then the constant remainder R is given by R = p (r).

Factor Theorem:

The number c is a solution of the polynomial equation p (x) = 0 if and only if p (x) has (x – c) as a factor.

Complete Factorization Theorem: If p (x) = an xn + an-1 xn-1 + . . . + a1 x + ao , an ≠ 0, then there are n numbers: c1 , c2 , . . . , cn , not necessarily distinct, such that p (x) = an (x – c1)(x – c2) . . . (x – cn) . The ci‘s are the zeros of p(x) and they may be either real or complex numbers. The complex zeros always occur in conjugate pairs whenever all coefficients of p (x) are real numbers. Fundamental Theorem of Algebra: Every nonconstant polynomial has at least one (real or complex) zero. Real Factors Theorem: Any polynomial with real coefficients can be factored into a product of linear and quadratic polynomials having real coefficients, where the quadratic polynomials have no real zeros. Rational Roots Theorem: Let an xn + an-1 xn-1 + . . . + a1 x + ao = 0 have all integral coefficients. If

c d

is a rational solution, in reduced form,

then c divides ao exactly and d divides an exactly. Descartes’s Rule of Signs: Suppose that p(x) is a polynomial with real coefficients and with terms written in descending powers of the variable. Then (i) the number of positive roots to p(x)=0 is either equal to N: the number of variations in sign in the coefficients of p(x), or else it is less than N by an even integer. (ii) the number of negative roots to p(x)=0 is either equal to M: the number of variations in sign in the coefficients of p(–x), or else it is less than M by an even integer. Upper and Lower Bound Theorem for Real Roots: Suppose the polynomial: p(x) = an xn + an-1 xn-1 + . . . + a1 x + ao has all real coefficients and an>0. Then (i) the positive number B is an upper bound on the real roots to p(x)=0 if p(x) = (x – B)(Bn-1 xn-1 + Bn-2 xn-2 + . . . + B1 x + Bo) + R1 and the numbers Bn-1 , Bn-2 , . . . , B1, Bo , R1 are all nonnegative. (ii) the negative number b is a lower bound on the real roots to p(x)=0 if p(x) = (x – b)(bn-1 xn-1 + bn-2 xn-2 + . . . + b1 x + bo) + R2 and the sequence of numbers bn-1 , bn-2 , . . . , b1, bo , R2 are alternately nonpositive and nonnegative or vice versa. Location Theorem:

If p(x) is a polynomial with real coefficients and a,b are real numbers with a < b and p(a) and p(b) have opposite sign then there exists at least one number c such that a < c < b and p(c) = 0....