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Square Roots and Radical Expressions
Solving Radical Equations
Simplifying Radical Expressions
Irrational Numbers in General and Square Roots in Particular
Roots of Polynomials
Simplifying Radical Expressions
Exponents and Radicals
Products and Quotients Involving Radicals
Roots of Quadratic Equations
Radical Expressions
Radicals and Rational Exponents
Find Square Roots and Compare Real Numbers
Radicals
Radicals and Rational Exponents
Theorems on the Roots of Polynomial Equations
SYNTHETIC DIVISION AND BOUNDS ON ROOTS
Simplifying Radical Expressions
Exponents and Radicals
Properties of Exponents and Square Roots
Solving Radical Equations
Rational Exponents and Radicals,Rationalizing Denominators
Rational Exponents and Radicals,Rationalizing Denominators
Quadratic Roots
Exponents and Roots
Multiplying Radical Expressions
Exponents and Radicals
Solving Radical Equations
Solving Quadratic Equations by Factoring and Extracting Roots
Newton's Method for Finding Roots
Roots of Quadratic Equations Studio
Roots, Radicals, and Root Functions
Review division factoring and Root Finding
Radicals
Simplifying Radical Expressions
Multiplying and Simplifying Radical Expressions
LIKE RADICALS
Multiplication and Division of Radicals
Radical Equations
BOUNDING ROOTS OF POLYNOMIALS
   
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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
then there are n numbers: not necessarily distinct,
such that

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

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:

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

and the numbers are all nonnegative.
(ii) the negative number b is a lower bound on the real roots to p(x)=0 if

and the sequence of numbers 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.

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