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**:

**Section 4.2**– Homogeneous Linear Equations: The General Solution

**Objective 10–1**: I can identify if two or more functions are linearly-independent.**Objective 10–2**: I can identify the characteristic equation for -degree homogeneous linear ODEs.**Objective 10–3**: I can find the most general solution to -degree homogeneous linear ODEs.

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

, where , has exactly solutions, counting multiplicities.

**Example 01**

**Example 02**

**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** have the form: .

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

**Example 06**

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, .

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, .

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

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

If , then the following are all equivalent:

- the columns of are linearly dependent vectors in
- the rows of are linearly dependent vectors in
- the matrix is not invertible
- the system of linear equations either has no solution or infinitely-many solutions

If , then the following are all equivalent:

- the columns of are linearly independent vectors in
- the rows of are linearly independent vectors in
- the matrix is invertible
- the system of linear equations has a unique solution

Let

Then,

Let

Then, using cofactors,

Find the determinants of the following matrices:

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