Algebr 3.4 notes

Chapter 3: Polynomial and Rational Functions

Section 3.4: Zeros of Polynomial Functions

Objectives

• Use the Rational Zero Theorem to find possible rational zeros.

• Find zeros of a polynomial function.

• Solve polynomial equations.

• Use the Linear Factorization Theorem to find polynomials with given zeros.

• Apply Descartes’s Rule of Signs.

The Rational Zero Theorem

• If a polynomial has integer coefficients and ( \frac{p}{q} ) (where ( \frac{p}{q} ) is in lowest terms) is a rational zero of f:

  • ( p ) is a factor of the constant term.

  • ( q ) is a factor of the leading coefficient.

Example 2: Using the Rational Zero Theorem

• Goal: List all possible rational zeros.

  • Constant term: [Insert value].

  • Leading coefficient: 4.

  • Factors of the constant term: [List factors].

  • Factors of the leading coefficient: 1, 2, 4.

  • Possible rational zeros: [Insert list].

Example 4: Finding Zeros of a Polynomial Function

Part 1: Listing Possible Rational Zeros

• Solution Steps:

  • List all possible rational zeros.

  • Apply synthetic division to test possible rational zeros.

Part 2: Testing Rational Zeros

• After testing, find a rational zero:

  • Successful rational zero: x = 2.

Part 3: Solving the Polynomial

• Find remaining zeros based on the identified rational zero.

• Solutions identified are: 2, [Insert other zeros if available].

Properties of Roots of Polynomial Equations

• A polynomial of degree n has n roots (counting multiplicities).

• If ( r ) is a root with real coefficients, then its conjugate ( ar{r} ) is also a root.

• Imaginary roots occur in conjugate pairs.

Example 5: Solving a Polynomial Equation

Part 1: Initial Steps

• List all possible rational roots.

• Apply synthetic division.

Part 2: Finding Roots

• Through synthetic division, identify x = 1 as a root.

• Factor the polynomial accordingly.

Part 3: Completing the Solution

• Continue to solve using the quadratic formula.

• Find the complete solution set of the polynomial equation.

The Fundamental Theorem of Algebra

• A polynomial of degree n has at least one complex root.

The Linear Factorization Theorem

• If ( f(x) ) is a polynomial, then it can be expressed as the product of linear factors:

  • ( f(x) = a(x - r_1)(x - r_2)...(x - r_n) ), where ( r_i ) are roots and ( a
    eq 0 ).

Example 6: Finding a Polynomial Function from Given Zeros

Part 1: Given Zeros

• Find a third-degree polynomial with real coefficients and known zeros, including i.

• By the conjugate root theorem, include -i.

Part 2: Application of Linear Factorization Theorem

• Using zeros: [Insert calculations and polynomial formed].

Descartes’ Rule of Signs

Positive Real Zeros

• The number of positive real zeros is determined by:

  • The number of sign changes in ( f(x) ).

  • Total possibilities: same as sign changes or less by a positive even integer.

Negative Real Zeros

• The number of negative real zeros is determined by:

  • The number of sign changes in ( f(-x) ).

  • Total possibilities: same as sign changes or less by a positive even integer.

Example 7: Using Descartes’ Rule of Signs

Part 1: Positive Real Zeros

• Counting sign changes in ( f(x) ): 4 changes.

  • Possible positive real zeros: 4, 2, or 0.

Part 2: Negative Real Zeros

• Counting sign changes in ( f(-x) ): 0 changes.

  • Conclusion: No negative real roots.