The Real Zeros of a Polynomial Function: Key Theorems and Methods

Section 4.6: The Real Zeros of a Polynomial Function

Remainder Theorem

The Remainder Theorem provides a method for finding the remainder when a polynomial f(x) is divided by a linear divisor of the form x - c. This theorem is useful for evaluating polynomials and for factoring.

• Statement: If a polynomial f(x) is divided by x - c, the remainder is f(c).

• Application: To find the remainder, substitute c into the polynomial.

• Example: To find the remainder when f(x) = x^3 + 3x^2 + 2x - 1 is divided by x + 2, compute f(-2).

Factor Theorem

The Factor Theorem is a special case of the Remainder Theorem and is fundamental for factoring polynomials and finding their zeros.

• Statement: x - c is a factor of f(x) if and only if f(c) = 0.

• Implication: If f(c) = 0, then c is a zero (root) of f(x).

• Example: To determine if x + 1 is a factor of f(x) = -2x^3 - x^2 + 4x + 3, check if f(-1) = 0.

Descartes' Rule of Signs

Descartes' Rule of Signs is a technique for determining the possible number of positive and negative real zeros of a polynomial function by analyzing the sign changes in its coefficients.

• Positive Real Zeros: The number of positive real zeros of f(x) is equal to the number of sign changes in the coefficients of f(x), or less than that by an even integer.

• Negative Real Zeros: The number of negative real zeros of f(x) is equal to the number of sign changes in the coefficients of f(-x), or less than that by an even integer.

• Example: For f(x) = x^4 + 2x^2 - 5x - 6, count the sign changes to estimate the number of positive and negative real zeros.

Rational Zeros Theorem

The Rational Zeros Theorem (also called Rational Root Theorem) provides a list of possible rational zeros for a polynomial with integer coefficients.

• Statement: For f(x) = a_nx^n + ... + a_0, any rational zero p/q (in lowest terms) must have p as a factor of the constant term a_0 and q as a factor of the leading coefficient a_n.

• Application: List all possible values of p/q and test them in the polynomial.

• Example: For f(x) = 3x^4 - 8x^3 - 7x^2 - 12, list all possible rational zeros using the Rational Zeros Theorem.

Steps for Finding the Real Zeros of a Polynomial Function

To systematically find the real zeros of a polynomial, follow these steps:

1. Step 1: Use the degree of the polynomial to determine the maximum number of real zeros.

2. Step 2: Use Descartes' Rule of Signs to determine the possible number of positive and negative real zeros.

3. Step 3:

   • If the polynomial has integer coefficients, use the Rational Zeros Theorem to list all possible rational zeros.

   • Test each possible zero using substitution, synthetic division, or long division. Each time a zero is found, factor it out and repeat the process with the reduced polynomial.

Bounds on Zeros Theorem

The Bounds on Zeros Theorem helps to determine intervals in which all real zeros of a polynomial must lie.

• If, in the process of dividing f(x) by x - M, all the coefficients in the result are positive (or all negative), then M is an upper (or lower) bound for the real zeros of f(x).

• Example: Use synthetic division to test bounds for the zeros of g(x) = x^4 - 2x^3 - 4x^2 + x - 1.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a polynomial function f(x) is continuous on an interval [a, b] and f(a) and f(b) have opposite signs, then there is at least one real zero between a and b.

• Application: This theorem is used to show the existence of a real zero in a given interval.

• Example: Show that f(x) = x^3 - x + 1 has a zero between -2 and -1 by evaluating f(-2) and f(-1).

Approximating the Real Zeros of a Polynomial Function

When exact zeros cannot be found algebraically, numerical methods can be used to approximate real zeros to a desired degree of accuracy.

1. Find two consecutive integers a and a+1 such that f(x) changes sign between them.

2. Divide the interval [a, a+1] into subintervals.

3. Evaluate f(x) at the endpoints of each subinterval to locate where the sign changes.

4. Continue subdividing the interval until the zero is approximated to the desired accuracy.

5. Example: Approximate the zero of f(x) = x^3 - x + 1 to two decimal places.

Summary Table: Key Theorems and Their Purposes

Theorem	Main Purpose
Remainder Theorem	Finds the remainder when dividing a polynomial by a linear divisor
Factor Theorem	Determines if a linear binomial is a factor of a polynomial
Descartes' Rule of Signs	Estimates the number of positive and negative real zeros
Rational Zeros Theorem	Lists all possible rational zeros of a polynomial
Bounds on Zeros Theorem	Finds upper and lower bounds for real zeros
Intermediate Value Theorem	Shows the existence of a real zero in an interval

Key Formulas and Notation

• Polynomial Division: $f(x) = (x - c)q(x) + r$

• Remainder Theorem: $r = f(c)$

• Rational Zeros Theorem: Possible zeros are $\pm \frac{p}{q}$, where $p$ divides $a_0$ and $q$ divides $a_n$

Example Problems

• Find the remainder: What is the remainder when $f(x) = x^3 + 3x^2 + 2x - 1$ is divided by $x + 2$? Solution: $f(-2) = (-2)^3 + 3(-2)^2 + 2(-2) - 1 = -8 + 12 - 4 - 1 = -1$

• List potential rational zeros: For $f(x) = 3x^4 - 8x^3 - 7x^2 - 12$, possible zeros are $\pm1, \pm2, \pm3, \pm4, \pm6, \pm12$ divided by $\pm1, \pm3$.

• Use Descartes' Rule of Signs: For $f(x) = x^4 + 2x^2 - 5x - 6$, count sign changes for $f(x)$ and $f(-x)$ to estimate zeros.