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Hawkes Learning Systems: College Algebra

Hawkes Learning Systems: College Algebra. Section 2.4: Higher Degree Polynomial Equations. Objectives. Solving quadratic-like equations. Solving general polynomial equations by factoring. Solving polynomial-like equations by factoring. Solving Quadratic-Like Equations.

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Hawkes Learning Systems: College Algebra

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  1. Hawkes Learning Systems: College Algebra Section 2.4: Higher Degree Polynomial Equations

  2. Objectives • Solving quadratic-like equations. • Solving general polynomial equations by factoring. • Solving polynomial-like equations by factoring.

  3. Solving Quadratic-Like Equations • A polynomial equation of degree in one variable, say , is an equation that can be written in the form where is a constant and . • In general, there is no method for solving polynomial equations that is guaranteed to find all solutions. • 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. 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. Example 1: Solving Quadratic-Like Equations Solve the quadratic-like equation. Step 1: Let and factor. Step 2: Replace with and solve for .

  6. Example 2: Solving Quadratic-Like Equations Solve the quadratic-like equation. or or or

  7. Solving General Polynomial Equations by Factoring • If an equation of the following type can be factored completely, then the equation can be solved by using the Zero-Factor Property. • If the coefficients in the polynomial are all real, the polynomial can, in principle, be factored. • 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. 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 or or

  9. Example 4: General Polynomial Equations Solve the equation by factoring.

  10. Example 5: General Polynomial Equations Solve the equation by factoring. or or or

  11. Solving Polynomial-Like Equations by Factoring • Some equations that are not polynomial can be solved using the methods we have developed in Section 2.4. • 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. 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. Example 7: Polynomial-Like Equations Solve the following equation by factoring. or or No Solution

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