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Exercises Polynomials and factoring

Example 7 from Algebra and Trigonometry, 5.3 Graphs of Polynomial Functions

Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function

Find the maximum number of turning points of each polynomial function.

ⓐ $f (x) = - x^{3} + 4 x^{5} - 3 x^{2} + 1$
ⓑ $f (x) = - {(x - 1)}^{2} (1 + 2 x^{2})$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. ⓐ
  First, rewrite the polynomial function in descending order: $f (x) = 4 x^{5} - x^{3} - 3 x^{2} + 1$
  
  Identify the degree of the polynomial function. This polynomial function is of degree 5.
  
  The maximum number of turning points is $5 - 1 = 4.$
2. ⓑ
  First, identify the leading term of the polynomial function if the function were expanded.
  
  Then, identify the degree of the polynomial function. This polynomial function is of degree 4.
  
  The maximum number of turning points is $4 - 1 = 3.$

How did it go?

PreviousIdentifying Zeros and Their Multiplicities Next Sketching the Graph of a Polynomial Function

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 5.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.