Theory of Equations: Finding Roots and Writing Polynomial Equations

Section 3.1: The Theory of Equations

Finding Roots of Polynomial Equations

This section explores how to determine the roots (solutions) of polynomial equations, including real and complex roots, and how to construct polynomial equations from given roots.

n-Root Theorem

• Statement: If $P(x) = 0$ is a polynomial equation with real or complex coefficients and positive degree n, then (when multiplicity is considered) $P(x) = 0$ has n roots.

• Multiplicity: A root has multiplicity m if it is repeated m times as a solution.

Key Terms and Definitions

• Root (Zero): A value of x for which $P(x) = 0$.

• Degree: The highest power of x in the polynomial.

• Multiplicity: The number of times a particular root occurs.

• Real Root: A root that is a real number.

• Complex Root: A root that is not real, typically involving the imaginary unit $i$ where $i^2 = -1$.

Examples: Finding Roots

• Example 1: $f(x) = x^4 - 8x + 16$

  • Degree: 4

  • Possible roots found by factoring or using the Rational Root Theorem.

  • Roots may be real or complex, and multiplicities are considered.

• Example 2: $6x^2 + 18x = 0$

  • Factor: $6x(x+3) = 0$

  • Roots: $x = 0$, $x = -3$

  • Degree: 2

• Example 3: $x^2 + 3 = 0$

  • Roots: $x^2 = -3 \implies x = \pm \sqrt{-3} = \pm i\sqrt{3}$

  • Complex roots occur in conjugate pairs.

Rational Root Theorem

• Possible rational roots of a polynomial equation $a_nx^n + ... + a_0 = 0$ are of the form $\pm \frac{p}{q}$, where p divides the constant term $a_0$ and q divides the leading coefficient $a_n$.

• Test each possible root by substitution or synthetic division.

Quadratic Formula

• For $ax^2 + bx + c = 0$, the roots are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• If $b^2 - 4ac < 0$, the roots are complex and occur in conjugate pairs.

Examples: Using the Quadratic Formula

• Example: $x^2 - 2x + 5 = 0$

  • $x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(5)}}{2}$

  • $x = \frac{2 \pm \sqrt{4 - 20}}{2}$

  • $x = \frac{2 \pm \sqrt{-16}}{2}$

  • $x = \frac{2 \pm 4i}{2}$

  • $x = 1 \pm 2i$

  • Imaginary roots occur in pairs: $1 + 2i$, $1 - 2i$

Writing Equations from Roots

Given a set of roots, you can construct a polynomial equation with real coefficients by multiplying factors of the form $(x - r)$ for each root $r$. If a root is complex or irrational, its conjugate must also be included to ensure real coefficients.

• Example: Roots at $x = 2$ and $x = -1 \pm i$

  • Factors: $(x - 2)$, $(x + 1 - i)$, $(x + 1 + i)$

  • Multiply conjugate pair: $(x + 1 - i)(x + 1 + i) = (x + 1)^2 - (i)^2 = (x + 1)^2 + 1$

  • Full equation: $y = (x - 2)[(x + 1)^2 + 1]$

  • Expand as needed for standard form.

• Example: Roots at $x = -6$, $x = 3 \pm i\sqrt{2}$

  • Factors: $(x + 6)$, $(x - 3 + i\sqrt{2})$, $(x - 3 - i\sqrt{2})$

  • Multiply conjugate pair: $(x - 3)^2 + 2$

  • Full equation: $y = (x + 6)[(x - 3)^2 + 2]$

  • Expand for standard form: $y = (x + 6)(x^2 - 6x + 11)$

Table: Types of Roots and Their Properties

Type of Root	Form	Occurs In	Notes
Real Root	r	All polynomials	Can be rational or irrational
Complex Root	a + bi	Polynomials with real coefficients	Always occur in conjugate pairs
Irrational Root	a + b\sqrt{c}	Polynomials with real coefficients	Always occur in conjugate pairs

Summary of Steps for Finding and Writing Roots

1. Determine the degree of the polynomial.

2. List all possible rational roots using the Rational Root Theorem.

3. Test possible roots by substitution or synthetic division.

4. Factor the polynomial as much as possible.

5. Use the quadratic formula for irreducible quadratics.

6. List all real and complex roots, including multiplicities.

7. To write a polynomial from given roots, include all conjugate pairs for complex or irrational roots and multiply the corresponding factors.

Additional info: The notes also include worked examples and graphical representations to illustrate the relationship between roots and the graph of a polynomial function.