Zeros of Polynomial Functions and Related Theorems

Zeros of Polynomial Functions

The Rational Zero Theorem

The Rational Zero Theorem provides a method for finding all possible rational zeros of a polynomial function with integer coefficients. It states that any rational zero, written as $\frac{p}{q}$ in lowest terms, must have a numerator $p$ that divides the constant term and a denominator $q$ that divides the leading coefficient.

• Possible Rational Zeros: For a polynomial $f(x) = a_n x^n + \ldots + a_0$, possible rational zeros are all values $\pm \frac{p}{q}$, where $p$ divides $a_0$ and $q$ divides $a_n$.

• Application: List all such values to test as potential zeros.

Example 1: List all possible rational zeros of $f(x) = 4x^5 + x^4 - 12x^3 - x$.

• Constant term $a_0 = -1$ (from $-x$), leading coefficient $a_5 = 4$.

• Possible values for $p$: divisors of $-1$ are $\pm1$.

• Possible values for $q$: divisors of $4$are$\pm1, \pm2, \pm4$.

• Possible rational zeros: $\pm1, \pm\frac{1}{2}, \pm\frac{1}{4}$.

Finding All Zeros of a Polynomial

To find all zeros of a polynomial, use the Rational Zero Theorem to list candidates, then use synthetic or long division to test each candidate. Continue factoring until all zeros are found.

Example 2: Find all zeros of $f(x) = 5x^3 + 2x^2 - x - 2$.

• List possible rational zeros using the Rational Zero Theorem.

• Test each candidate by substitution or synthetic division.

• Factor further as needed to find all real and complex zeros.

Properties of Roots of Polynomial Equations

• Number of Roots: A polynomial equation of degree $n$ has exactly $n$ roots (counting multiplicities).

• Complex Conjugate Root Theorem: If $a + bi$ is a root of a polynomial with real coefficients, then $a - bi$ is also a root. Thus, non-real roots occur in conjugate pairs.

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-zero, single-variable polynomial of degree $n$ with complex coefficients has exactly $n$ roots in the complex number system (counting multiplicities).

The Linear Factorization Theorem

This theorem states that any polynomial of degree $n$ with complex coefficients can be factored into $n$ linear factors:

$P(x) = a_n(x - r_1)(x - r_2)\ldots(x - r_n)$

where $r_1, r_2, \ldots, r_n$ are the roots of the polynomial (real or complex).

Example 3: Solving a Polynomial Equation

Solve $6x^4 - 22x^3 + 30x^2 - 13x + 0 = 0$.

• Apply the Rational Zero Theorem to list possible rational zeros.

• Test candidates and factor the polynomial as needed.

• Continue until all zeros are found.

Example 4: Constructing a Polynomial from Given Zeros

Find a third-degree polynomial function $f(x)$ with real coefficients that has $-3$ and $i$ as zeros and such that $f(1) = 8$.

• Since coefficients are real, the conjugate $-i$ must also be a zero.

• The zeros are $-3, i, -i$.

• Form the polynomial: $f(x) = a(x + 3)(x - i)(x + i) = a(x + 3)(x^2 + 1)$.

• Use $f(1) = 8$ to solve for $a$:

• $f(1) = a(1 + 3)((1)^2 + 1) = a(4)(2) = 8a$

• So $8a = 8 \implies a = 1$.

• Final answer: $f(x) = (x + 3)(x^2 + 1)$.

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.

• 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 number.

• 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 number.

Example 5: Applying Descartes’ Rule of Signs

Determine the possible number of positive and negative real zeros of $f(x) = 14x^4 - 71x^3 + 154x^2 - 120x + 0$.

• Count sign changes in $f(x)$ for positive zeros.

• Substitute $-x$ for $x$ and count sign changes in $f(-x)$ for negative zeros.

• The actual number of positive or negative real zeros is equal to the number of sign changes, or less than that by an even number.