Hawkes Learning Systems: College Algebra

1. Hawkes Learning Systems:College Algebra Section 5.1: Introduction to Polynomial Equations and Graphs

2. Objectives • Zeros of polynomials and solutions of polynomial equations. • Graphing factored polynomials. • Solving polynomial inequalities.

3. Zeros of a Polynomial The number k is said to be a zero of the polynomial function if . This is also expressed by saying that k is a root or a solution of the equation Note: k may be a complex number.

4. Zeros of a Polynomial If f is a polynomial with real coefficients and if k is a real number zero of f, then the statement means the graph of f crosses the x-axis at In this case, may be referred to as an x-intercept of f. .

5. Polynomial Equations A polynomial equation in one variable, say the variable x, is an equation that can be written in the form where are constants. Assuming , we say such an equation is of degreen. For example:

6. Example: Zeros of Polynomials and Solutions of Polynomial Equations Verify that the given value of solves the corresponding polynomial equation. Substitute –2 for x in the original equation. Simplify, and solve the equation. Thus, is a solution to the equation.

7. Graphing Factored Polynomials The behavior of a polynomial function as can be determined as follows: • As , the leading term of . dominates the behavior. • If n is even, as , and if n is odd, then . as and as . • If an is positive, multiplying by an merely compresses or stretches the graph of , while if an is negative, the graph of is the reflection with respect to the x-axis of the graph of .

8. Graphing Factored Polynomials End Behavior of a Polynomial Given a polynomial we have

9. Graphing Factored Polynomials End Behavior of a Polynomial (cont.)

10. Graphing Factored Polynomials Summary: Note: stretches or compresses the graph.

11. Graphing Factored Polynomials For the y-intercept is

12. Graphing Factored Polynomials If we are able to factor a given polynomial f into a product of linear factors, every linear factor with real coefficients will correspond to an x-intercept of the graph of f. For example, has the x-intercepts:

13. Example: Graphing Factored Polynomials Sketch the graph of the following polynomial function, paying particular attention to the x-intercept(s), the y-intercept, and the behavior as

14. Example: Graphing Factored Polynomials Sketch the graph of the following polynomial function, paying particular attention to the x-intercept(s), the y-intercept, and the behavior as If we were to multiply out the three linear factors of f, the highest degree term would be . The degree of f and the fact that the leading coefficient is negative indicates how f behaves as

15. Solving Polynomial Inequalities Every polynomial inequality can be rewritten in the form where f is a polynomial function. This will be the key to solving the inequality. By graphing the polynomial f, we will be able to easily pick out the intervals that solve the inequality.

16. Example: Solving Polynomial Inequalities Solve the following polynomial inequality. Now graph the function using:

17. Example: Solving Polynomial Inequalities Solve the following polynomial inequality. Now graph the function using:

18. Example: Solving Polynomial Inequalities Solve the following polynomial inequality. Graph the function using: