1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 5.1: Introduction to Polynomial Equations and Graphs

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Objectives o Zeros of polynomials and solutions of polynomial equations. o Graphing factored polynomials. o Solving polynomial inequalities.

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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 degree n. For example:

6. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 1: 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. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 2: Zeros of Polynomials and Solutions of Polynomial Equations Verify that the given value of solves the corresponding polynomial equation. Although we could verify the solution by substituting for x, it is easier to solve this equation for ourselves using the quadratic formula. Continued on the next slide…

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 2: Zeros of Polynomials and Solutions of Polynomial Equations (Cont.) One of the two resulting solutions for x is equivalent to the value we were given for x at the beginning of the problem, and thus the given value of x solves the equation.

9. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 3: Zeros of Polynomials and Solutions of Polynomial Equations Verify that the given value of x solves the corresponding polynomial equation.

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Graphing Factored Polynomials The behavior of a polynomial function as can be determined as follows: o As, the leading term of. dominates the behavior. o If n is even, as, and if n is odd, then. as and as. o If a n is positive, multiplying by a n merely compresses or stretches the graph of, while if a n is negative, the graph of is the reflection with respect to the x -axis of the graph of.

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Graphing Factored Polynomials Summary: n even n odd No change. is reflected over the x -axis. Note: stretches or compresses the graph.

12. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Graphing Factored Polynomials For the y -intercept is

13. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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:

14. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 4: 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. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 5: 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

16. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 6: 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

17. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. 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.

18. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 7: Solving Polynomial Inequalities Solve the following polynomial inequality. Now graph the function using:

19. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 8: Solving Polynomial Inequalities Solve the following polynomial inequality. Now graph the function using:

20. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 9: Solving Polynomial Inequalities Solve the following polynomial inequality. Graph the function using: