Zeros of Polynomial Functions (Section 3.3) – Study Notes

Zeros of Polynomial Functions

Introduction

Understanding the zeros of polynomial functions is essential in algebra. This section covers theorems and procedures for finding zeros, factoring polynomials, and analyzing their structure.

Theorem 1: Factor Theorem

Statement and Application

• Factor Theorem: For any polynomial function f(x), (x - c) is a factor of the polynomial if and only if f(c) = 0.

• This theorem provides a direct connection between the zeros of a polynomial and its factors.

Example 1: Let $f(x) = x^2 - 6x - 7$.

• Find $f(7)$ and $f(-1)$:

  • $f(7) = 7^2 - 6 \times 7 - 7 = 49 - 42 - 7 = 0$

  • $f(-1) = (-1)^2 - 6 \times (-1) - 7 = 1 + 6 - 7 = 0$

• Since $f(7) = 0$ and $f(-1) = 0$, both $(x - 7)$ and $(x + 1)$ are factors of $f(x)$.

Example 2: Factor $f(x) = 6x^3 + 19x^2 + 2x - 3$ into linear factors, given that $-3$ is a zero.

• Since $f(-3) = 0$, $(x + 3)$ is a factor. Use polynomial or synthetic division to factor further.

Theorem 2: Rational Zeros Theorem

Statement and Procedure

• Rational Zeros Theorem: If $\frac{p}{q}$ is a rational zero of a polynomial function with integer coefficients, then:

  • p is a factor of the constant term.

  • q is a factor of the leading coefficient.

Procedure: Finding Rational Zeros of a Polynomial

1. List all possible rational zeros: Use the Rational Zeros Theorem to list all possible values of $\frac{p}{q}$.

2. Test each candidate: Substitute each candidate into the polynomial (or use synthetic division) to check if it is a zero (i.e., if the remainder is zero).

3. Repeat for the quotient: Once a zero is found, factor it out and repeat the process for the reduced polynomial until all zeros are found or the remaining polynomial factors easily.

Example 3: Consider $f(x) = 6x^4 + 7x^3 - 12x^2 - 3x + 2$.

• (a) List all possible rational zeros:

  • Possible values for $p$: factors of 2 ($\pm1, \pm2$)

  • Possible values for $q$: factors of 6 ($\pm1, \pm2, \pm3, \pm6$)

  • Possible rational zeros: $\pm1, \pm2, \pm\frac{1}{2}, \pm\frac{1}{3}, \pm\frac{1}{6}, \pm\frac{2}{3}, \pm\frac{2}{6}$ (simplify as needed)

• (b) Test each candidate and factor $f(x)$ into linear factors.

Theorem 3: Fundamental Theorem of Algebra

Statement

• Every function defined by a polynomial of degree 1 or more has at least one complex zero.

Fact 1: Number of Zeros Theorem

• A polynomial of degree $n$ has at most $n$ distinct zeros (real or complex).

Theorem 4: Complete Factorization Theorem

Statement

• If $P(x)$ is a polynomial of degree $n \geq 1$, then there exist complex numbers $a, c_1, c_2, \ldots, c_n$ (with $a \neq 0$) such that:

$P(x) = a(x - c_1)(x - c_2)\cdots(x - c_n)$

• Here, $a$ is the leading coefficient of $P(x)$.

Definition: Multiplicity of a Zero

• The multiplicity of a zero refers to the number of times a specific zero (or its corresponding linear factor) appears in the completely factored form of a polynomial.

Examples: Constructing Polynomials from Zeros

• Example 4(a): Find a degree 3 polynomial with zeros at $-1$, $2$, and $4$, and $f(1) = 3$.

  • General form: $f(x) = a(x + 1)(x - 2)(x - 4)$

  • Plug in $x = 1$ to solve for $a$:

  • $f(1) = a(1 + 1)(1 - 2)(1 - 4) = a(2)(-1)(-3) = a \times 6$

  • Set $a \times 6 = 3 \implies a = \frac{1}{2}$

  • So, $f(x) = \frac{1}{2}(x + 1)(x - 2)(x - 4)$

• Example 4(b): Find a degree 3 polynomial where $-2$ is a zero of multiplicity 3 and $f(-1) = 4$.

  • General form: $f(x) = a(x + 2)^3$

  • Plug in $x = -1$ to solve for $a$:

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

  • Set $a = 4$

  • So, $f(x) = 4(x + 2)^3$