1. Chapter 5 Polynomials and Polynomial Functions © Tentinger

2. Essential Understanding and Objectives Essential Understanding: the factors of a n and a 0 in P(x) can help you factor P(x) and solve the equation P(x) = 0 Objectives Students will be able to: Solve equations using the Rational Root Theorem Use the Conjugate Root Theorem

3. Iowa Core Curriculum Number and Quantity N.CN.7. Solve quadratic equations with real coefficients that have complex solutions N.CN.8. (+) Extend polynomial identities to the complex numbers

4. Rational Root Theorem There are a limited number of possible roots of P(x) = 0 Integer roots must be factors of a 0 Rational roots must have reduced form p/q where p is an integer factor of a 0 (constant) and q is an integer factor of a n (leading coefficient). 21x 2 + 29x + 10 = 0 Factors of leading coefficient: 1, 3, 7, 21 Factors of Constant Term: 1, 2, 5, 10 Divide everything by leading coefficient

5. Example What are the rational roots of 2x 3 – x 2 + 2x + 5 = 0 Plus or minus: 1, 5, ½, 5/2 Make a table, see where the y value will be zero What are the rational roots of 3x 3 + 7x 2 + 6x – 8 = 0 What are the rational roots of 2x 3 +x 2 – 7x – 6 = 0 Go through and test for each root, then you can factor to find other roots.

6. Conjugate Root Theorem If P(x) is a polynomial with rational coefficients, then irrational roots of P(x) = 0 that have the form a + √b is an irrational root with a and b rational, then a - √b is also a root If a + √b is a root then, a - √b is a root If a +bi is a root then, a – bi is a root A cubic polynomial has real coefficients. If 3 – 2i and 5/2 are two roots, what is one additional root? Constructing a polynomial function What quartic polynomial equation has roots 2 – 3i, 8, 2?

7. Descartes Rule of Signs The number of positive real roots of P(x) = 0 is either equal to the number of sign changes between consecutive coefficients of P(x) or is less that by an even number The number of negative real roots of P(x) = 0 is either equal to the number of sign changes between consecutive coefficient of P(-x) or is less than that by an even number x 3 – x 2 + 1 = 0 2x 4 – x 3 + 3x 2 – 1 = 0