The Real Zeros of a Polynomial Function

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Presentation transcript:

The Real Zeros of a Polynomial Function Section 4.2 The Real Zeros of a Polynomial Function Copyright © 2013 Pearson Education, Inc. All rights reserved

Use the Remainder and Factor Theorems. Objectives Use the Remainder and Factor Theorems. Use the Rational Zeros theorem to list the potential rational zeros of a polynomial function. Find the real zeros of a polynomial function. Solve polynomial equations Use the theorem for bounds on zeros. Use the Intermediate Value Theorem. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

The remainder is –1. The remainder is 5. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

By the factor theorem, x + 1 is a factor of f(x). By the factor theorem, x – 1 is not a factor of f(x). Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Factors of the constant Factors of the leading coefficient Copyright © 2013 Pearson Education, Inc. All rights reserved

Since f is a polynomial of degree 3, there are at most three real zeros. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Step 1: There are at most 4 real zeros. Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

BOUND Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Copyright © 2013 Pearson Education, Inc. All rights reserved

Homework: 4.2 page 209 #11, 13, 21, 27, 33, 39, 45, 47, 53, 63, 71, 75