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Exercises Polynomials and factoring

Example 9 from Algebra and Trigonometry, 5.3 Graphs of Polynomial Functions

Using the Intermediate Value Theorem

Show that the function $f (x) = x^{3} - 5 x^{2} + 3 x + 6$ has at least two real zeros between $f (x)$ and $x = 4.$

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.

Solution
As a start, evaluate $f (x)$ at the integer values $x = 1, 2, 3,$ and $4.$ See Table 2.

$x$ 1 2 3 4

$f (x)$ 5 0 –3 2

Table 2

We see that one zero occurs at $x = 2.$ Also, since $f (3)$ is negative and $f (4)$ is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4.
We have shown that there are at least two real zeros between $x = 1$ and $x = 4.$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Show that the function $f (x) = 7 x^{5} - 9 x^{4} - x^{2}$ has at least one real zero between $x = 1$ and $x = 2.$

How did it go?

PreviousSketching the Graph of a Polynomial Function Next Writing a Formula for a Polynomial Function from the Graph

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.