Algebra 3.3 notes

College Algebra Overview

• Textbook: College Algebra Eighth Edition

• Chapter: 3 - Polynomial and Rational Functions

Section 3.3: Dividing Polynomials; Remainder and Factor Theorems

• Key Concepts in Section 3.3:

  • Division of polynomials using long division and synthetic division.

  • The Remainder Theorem for polynomial evaluation.

  • The Factor Theorem for solving polynomial equations.

Objectives

• Long Division: Understand how to divide polynomials using long division.

• Synthetic Division: Learn the method for dividing polynomials using synthetic division.

• Remainder Theorem: Evaluate a polynomial accurately utilizing the Remainder Theorem.

• Factor Theorem: Apply the Factor Theorem for solving polynomial equations efficiently.

Long Division of Polynomials

Steps for Long Division

• Step 1: Arrange dividend and divisor in descending powers of variables.

• Step 2: Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.

• Step 3: Multiply the entire divisor by this new quotient term.

• Step 4: Write this product beneath the dividend, aligning like terms.

• Step 5: Subtract to find the new remainder.

• Step 6: Bring down the next term from the dividend to form a new divided expression. Repeat the process until the degree of the remainder is less than that of the divisor.

The Division Algorithm

• Definition: For polynomials, if dividend and divisor are denoted as P(x) and D(x) respectively, where the degree of D(x) is less than or equal to the degree of P(x), there will exist unique polynomials Q(x) and R(x) such that:

  • P(x) = D(x) * Q(x) + R(x)

• Properties:

  • R(x) = 0 or deg(R) < deg(D)

  • If R(x) = 0, it indicates that D(x) divides evenly into P(x).

Synthetic Division

Steps for Synthetic Division

• Step 1: Arrange the polynomial in descending order and ensure all coefficients are represented, filling in with zeros where necessary.

• Step 2: Identify the divisor, usually formatted as (x - c).

• Step 3: Write the coefficients under the division format and bring down the leading coefficient.

• Step 4: Multiply c by the value on the bottom row and write in the column above.

• Step 5: Sum the columns to create the new row below.

• Step 6: Continue until all coefficients are addressed. The result gives you the quotient and the remainder.

The Remainder Theorem

• Statement: When dividing a polynomial P(x) by (x - c), the remainder is P(c).

• Application: Useful for evaluating polynomials quickly.

Factor Theorem

• Definition: For a polynomial P(x) and constant a:

  • If P(a) = 0, then (x - a) is a factor of P(x).

  • Conversely, if (x - a) is a factor of P(x), then P(a) = 0.

Examples

Example 2: Long Division of Polynomials

• Follow the specific steps laid out in previous sections for dividing the given polynomial.

Example 4: Using Synthetic Division

• Adhere to structured division, ensuring coefficients are correctly represented, and finalize with the result of synthetic division.

Example 5: Remainder Theorem Application

• Dividing the polynomial and evaluating using synthetic division to find relevant outputs.

Example 6: Factor Theorem in Action

• Demonstrate the application of the Factor Theorem to derive factors from given polynomial zeros and combine it with synthetic division to simplify the expression.