Homogeneous Equations

Homogeneous EquationsExpected Educational ResultsAlgebraFundamental Theorem of AlgebraTheorem: Fundamental Theorem of AlgebraCorollary: Fundamental Theorem of AlgebraSolving PolynomialsDistributive Property and Zero Product Property of Real NumbersQuadratic EquationsQuadratic-Like EquationsDefinition: Quadratic-Like EquationMethod to Solve Quadratic-Like EquationsFinding Zeros Using Rational Root Theorem and Polynomial or Synthetic DivisionFinding Zeros Using Synthetic DivisionPractice 01DeterminantsProperties of DeterminantsComputation of Determinants Matrix MatrixPractice 02CC BY-NC-SA 4.0

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Algebra

Fundamental Theorem of Algebra

Theorem: Fundamental Theorem of Algebra

A polynomial of degree , , , has exactly factors, counting multiplicities.

Corollary: Fundamental Theorem of Algebra

, where , has exactly solutions, counting multiplicities.

Solving Polynomials

Distributive Property and Zero Product Property of Real Numbers

Example 01

Example 02

Quadratic Equations

Example 03

Distributive Property and Zero Product Property of Real Numbers

Example 04

Distributive Property and Zero Product Property of Real Numbers

Example 05

Quadratic Formula

Quadratic-Like Equations

Definition: Quadratic-Like Equation

Quadratic-like equations have the form: .

Method to Solve Quadratic-Like Equations

  1. Let and substitute into the quadratic-like equation;
  2. Factor: , where \text{ and } ;
  3. Solve in terms of : ;
  4. Solve in terms of :

Example 06

Finding Zeros Using Rational Root Theorem and Polynomial or Synthetic Division

Given the polynomial

Let be the factors of . Let be the factors of . Then the set of potential rational roots of the polynomial are

Example 07

Find all three real roots of: .

The factors of are .

The factors of are .

Then the set of distinct potential rational roots of the polynomial are:

Now use polynomial or synthetic division to find the three factors.

Using Polynomial Division

a. Divide the polynomial by , i.e., divide by :

Since the above polynomial division has a non-zero remainder, then is not a factor of .

b. Divide the polynomial by , i.e., divide by :

Since the above polynomial division has a zero remainder, then is a factor of .

c. Divide the polynomial by , i.e., divide by :

Since the above polynomial division has a zero remainder, then is a factor of .

d. Divide the polynomial by , i.e., divide by :

Since the above polynomial division has a zero remainder, then is a factor of .

Thus, .

Finding Zeros Using Synthetic Division

a. Divide the polynomial by , i.e., divide by :

The steps above are provided to show the method of synthetic division. Your work should only look like the following:

Since the above polynomial division has a non-zero remainder (the right-most number on the last row), then is not a factor of .

b. Divide the polynomial by , i.e., divide by :

Since the above polynomial division has a zero remainder (the right-most number on the last row), then is a factor of .

c. Divide the polyomial by , i.e., divide by :

Since the above polynomial division has a zero remainder (the right-most number on the last row), then is a factor of .

d. Divide the polyomial by , i.e., divide by :

Since the above polynomial division has a zero remainder (the right-most number on the last row), then is a factor of .

Thus, .

Practice 01

Solve, i.e., find all solutions, to the following equations:

Determinants

Let be a square matrix. The determinant of is represented by det() or .

Properties of Determinants

If , then the following are all equivalent:

If , then the following are all equivalent:

Computation of Determinants

Matrix

Let

Then,

Matrix

Let

Then, using cofactors,

Practice 02

Find the determinants of the following matrices:

CC BY-NC-SA 4.0

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Last Modified: Monday, 6 September 2020 13:33 EDT