Polynomial Functions and Roots: College Algebra Study Notes

Polynomial Functions

Introduction to Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. They are fundamental in College Algebra for modeling various types of relationships and solving equations.

• Definition: A polynomial function of degree n has the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where $a_n \neq 0$.

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

• Roots/Zeros: Values of x for which $f(x) = 0$.

• Factoring: Expressing a polynomial as a product of lower-degree polynomials.

Finding Roots of Polynomial Functions

Roots (or zeros) of a polynomial are the solutions to the equation $f(x) = 0$. Several algebraic techniques are used to find these roots, including factoring, synthetic division, and the Rational Root Theorem.

• Factoring: If a polynomial can be factored, set each factor equal to zero and solve for x.

• Synthetic Division: A shortcut method for dividing a polynomial by a binomial of the form $(x - c)$.

• Rational Root Theorem: Provides a list of possible rational roots based on the factors of the constant term and leading coefficient.

• Quadratic Formula: For degree 2 polynomials: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Find the roots of $f(x) = x^3 - 6x^2 + 11x - 6$.

• Possible rational roots: ±1, ±2, ±3, ±6 (from Rational Root Theorem).

• Test $x = 1$: $f(1) = 1 - 6 + 11 - 6 = 0$ (so $x = 1$ is a root).

• Divide $f(x)$ by $(x - 1)$ using synthetic division to find the remaining quadratic factor.

• Factor the quadratic: $x^2 - 5x + 6 = (x - 2)(x - 3)$, so roots are $x = 2$ and $x = 3$.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a binomial of the form $(x - c)$. It is especially useful for finding roots and factoring polynomials.

• Steps:

  1. Write the coefficients of the polynomial.

  2. Write the root c to the left.

  3. Bring down the leading coefficient.

  4. Multiply c by the number just written below, add to the next coefficient, and repeat.

• Result: The final row gives the coefficients of the quotient polynomial and the remainder.

Example: Divide $f(x) = x^3 - 6x^2 + 11x - 6$ by $(x - 1)$ using synthetic division.

• Coefficients: 1, -6, 11, -6

• c = 1

• Process yields: 1, -5, 6, 0 (so quotient is $x^2 - 5x + 6$)

Quadratic Equations and Roots

Once a cubic or higher-degree polynomial is reduced to a quadratic, the roots can be found using factoring or the quadratic formula.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Factoring: Express $ax^2 + bx + c$ as $(x - r_1)(x - r_2)$ where $r_1$ and $r_2$ are roots.

Example: $x^2 - 5x + 6 = 0$ factors to $(x - 2)(x - 3) = 0$, so $x = 2$ and $x = 3$.

Summary Table: Methods for Finding Polynomial Roots

Method	When to Use	Steps
Factoring	Simple polynomials, recognizable patterns	Factor, set each factor to zero, solve for x
Synthetic Division	Testing possible roots, dividing by (x - c)	Use coefficients, test root, obtain quotient and remainder
Quadratic Formula	Quadratic polynomials	Plug into $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Rational Root Theorem	Higher-degree polynomials	List possible rational roots, test each

Additional info:

• Some questions on the sheet involve finding all roots of a cubic polynomial, using synthetic division, and applying the quadratic formula.

• These methods are essential for solving polynomial equations in College Algebra.