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 , where .

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

• Roots/Zeros: Values of x for which .

• 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 . 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 .

• 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:

Example: Find the roots of .

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

• Test : (so is a root).

• Divide by using synthetic division to find the remaining quadratic factor.

• Factor the quadratic: , so roots are and .

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a binomial of the form . 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 by using synthetic division.

• Coefficients: 1, -6, 11, -6

• c = 1

• Process yields: 1, -5, 6, 0 (so quotient is )

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:

• Factoring: Express as where and are roots.

Example: factors to , so and .

Summary Table: Methods for Finding Polynomial Roots

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.