1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 2.4: Higher Degree Polynomial Equations

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Objectives o Solving quadratic-like equations. o Solving general polynomial equations by factoring. o Solving polynomial-like equations by factoring.

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Quadratic-Like Equations o A polynomial equation of degree in one variable, say, is an equation that can be written in the form where is a constant and. o In general, there is no method for solving polynomial equations that is guaranteed to find all solutions. o Since the Zero-Factor property applies whenever a product of any finite number of factors is equal to 0, we can use this property to solve quadratic-like equations.

4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Quadratic-Like Equations An equation is quadratic-like, or quadratic in form, if it can be written in the form Where,, and are constants,, and is an algebraic expression. Such equations can be solved by first solving for and then solving for the variable in the expression.

5. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 1: Solving Quadratic-Like Equations Solve the quadratic-like equation. Step 1: Let and factor. Step 2: Replace with and solve for.

6. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 2: Solving Quadratic-Like Equations Solve the quadratic-like equation. or

7. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving General Polynomial Equations by Factoring o If an equation of the following type can be factored completely, then the equation can be solved by using the Zero-Factor Property. o If the coefficients in the polynomial are all real, the polynomial can, in principle, be factored. o In practice, this may be difficult to accomplish unless the degree of the polynomial is small or the polynomial is easily recognizable as a special product.

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 3: General Polynomial Equations Solve the equation by factoring. Step 1: Isolate on one side and factor. Step 2: Set both equations equal to and solve. or

9. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 4: General Polynomial Equations Solve the equation by factoring.

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 5: General Polynomial Equations Solve the equation by factoring. or

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Polynomial-Like Equations by Factoring o Some equations that are not polynomial can be solved using the methods we have developed in Section 2.4. o The general idea will be to rewrite the equation so that 0 appears on one side, and then to apply the Zero-Factor Property.

12. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 6: Polynomial-Like Equations Solve the following equation by factoring. Step 1: Isolate on one side. Step 2: Factor. Step 3: Apply the Zero-Factor Property.

13. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 7: Polynomial-Like Equations Solve the following equation by factoring. No Solution or