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Real Zeros of Polynomials

Remainder, Factor, and Rational Zero Theorems and Descartes' Rule of Signs.

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Real Zeros of Polynomials

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Descartes' Rule of Signs

Descartes' rule of signs is a technique for determining the number of positive and negative real roots of a polynomial.

factor theorem

The factor theorem states that if f(x) is a polynomial of degree n>0 and f(c)=0, then x-c is a factor of the polynomial f(x).

factorization theorem

The factorization theorem states that If f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}, where a_{n} \ne 0, and n is a positive integer, then f(x)=a_{n}(x-c_{1})(x-c_{2})\cdots(x-c_{0}) where the numbers c_{i} are complex numbers.


The multiplicity of a term describes the number of times the given term acts as a zero of the given function.


A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.

Rational Zero Theorem

The rational zero theorem states that for a polynomial, f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0, where a_n, a_{n-1}, \cdots a_0 are integers, the rational roots can be determined from the factors of a_n and a_0. More specifically, if p is a factor of a_0 and q is a factor of a_n, then all the rational factors will have the form \pm \frac{p}{q}.

Remainder Theorem

The remainder theorem states that if f(k) = r, then r is the remainder when dividing f(x) by (x - k).


The roots of a function are the values of x that make y equal to zero.

Synthetic Division

Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.


The zeroes of a function f(x) are the values of x that cause f(x) to be equal to zero.


The zeros of a function f(x) are the values of x that cause f(x) to be equal to zero.

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