Chapter 3 – Polynomial and Rational Functions 3.4 - Real Zeros of Polynomials.

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Presentation transcript:

Chapter 3 – Polynomial and Rational Functions Real Zeros of Polynomials

Example A polynomial in factored form: A polynomial in expanded form: Real Zeros of Polynomials

Theorem Real Zeros of Polynomials

Finding the Rational Zeros Real Zeros of Polynomials

Descartes'’ Rule of Signs To understand this rule we need to understand the concept of variation in sign. If P(x) is a polynomial with real coefficients, written with descending powers of x and excluding powers with a 0 coefficient, then a variation of sign occurs whenever adjacent coefficients have opposite signs Real Zeros of Polynomials

Descartes’ Rule of Signs Real Zeros of Polynomials

Example This polynomial has 3 variations in sign meaning P(x) has either 3 or 1 positive zeros. P(-x) = -5x 7 + 3x 5 – x 4 + 2x 2 – x – 3 has 4 variations in sign meaning P(x) has either 4 or 2 negative zeros Real Zeros of Polynomials

Examples – pg. 260 Find all rational zeros of the polynomial and write the polynomial in factored form Real Zeros of Polynomials

Examples – pg. 261 Find all real zeros of the polynomial. Use the quadratic formula if necessary Real Zeros of Polynomials

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