The Factor Theorem tells us that finding the zeros of a polynomial is the same thing as factoring it into linear factors

 Rational Zeros of Polynomials

From the factored form we see that the zeros of P are 2, 3, and -4. When the polynomial is expanded, the constant 24 is obtained by multiplying (-2) X (-3) X 4. This means that the zeros of the polynomial are all factors of the constant term. The following generalizes this observation.

Example of Using the Rational Zeros Theorem

Example of Using the Rational Zeros Theorem and the Quadratic Formula

Descartes’ Rule of Signs

If P(x) is a polynomial with real coefficients, written with descending powers of x (and omitting powers with coefficient 0), then a variation in sign,/b> occurs whenever adjacent coefficients have opposite signs

Example of Using Descartes’ Rule

 Upper and Lower Bounds Theorem

Example of Upper and Lower Bounds for the Zeros of a Polynomial

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Practice