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

2. Example A polynomial in factored form: A polynomial in expanded form: 3.4 - Real Zeros of Polynomials

3. Theorem 3.4 - Real Zeros of Polynomials

4. Finding the Rational Zeros 3.4 - Real Zeros of Polynomials

5. 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. 3.4 - Real Zeros of Polynomials

6. Descartes’ Rule of Signs 3.4 - Real Zeros of Polynomials

7. 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. 3.4 - Real Zeros of Polynomials

8. Examples – pg. 260 Find all rational zeros of the polynomial and write the polynomial in factored form. 3.4 - Real Zeros of Polynomials

9. Examples – pg. 261 Find all real zeros of the polynomial. Use the quadratic formula if necessary. 3.4 - Real Zeros of Polynomials