chapter 3 – polynomial and rational functions 3.4 - real zeros of polynomials
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3.4 - Real Zeros of Polynomials
Section 3.4 Real Zeros of Polynomials
Chapter 3 – Polynomial and Rational Functions
3.4 - Real Zeros of Polynomials
ExampleA polynomial in factored form:
A polynomial in expanded form:
2 3 4P x x x x
3 2 14 24P x x x x
3.4 - Real Zeros of Polynomials
Theorem
3.4 - Real Zeros of Polynomials
Finding the Rational Zeros
3.4 - 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.
3.4 - Real Zeros of Polynomials
Descartes’ Rule of Signs
3.4 - Real Zeros of Polynomials
Example
This polynomial has 3 variations in sign meaning P(x) has either 3 or 1 positive zeros.
P(-x) = -5x7 + 3x5 – x4 + 2x2 – x – 3 has 4 variations in sign meaning P(x) has either 4 or 2 negative zeros.
3.4 - Real Zeros of Polynomials
Examples – pg. 260Find all rational zeros of the polynomial and write
the polynomial in factored form.
3 2
4 2
5 4 3 2
6 5 4 3 2
16. 7 14 8
25. 5 4
43. 3 9 31 36
46. 2 3 13 29 27 32 12
P x x x x
P x x x
P x x x x x
P x x x x x x x
3.4 - Real Zeros of Polynomials
Examples – pg. 261Find all real zeros of the polynomial. Use the
quadratic formula if necessary.
3 2
3 2
5 4 3 2
48. 5 2 12
53. 4 6 1
56. 4 18 6 91 60 9
P x x x x
P x x x
P x x x x x x