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Exercises Polynomials and factoring

Example 9 from Algebra and Trigonometry, 5.2 Power Functions and Polynomial Functions

Determining the Intercepts of a Polynomial Function with Factoring

Given the polynomial function $f (x) = x^{4} - 4 x^{2} - 45,$ determine the y- and x-intercepts.

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
The y-intercept occurs when the input is zero.

$\begin{matrix} f (0) & = & {(0)}^{4} - 4 {(0)}^{2} - 45 \\ = & −45 \end{matrix}$
The y-intercept is $(0, −45) .$
The x-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.

$\begin{matrix} f (x) & = & x^{4} - 4 x^{2} - 45 \\ = & (x^{2} - 9) (x^{2} + 5) \\ = & (x - 3) (x + 3) (x^{2} + 5) \end{matrix}$
$0 = (x - 3) (x + 3) (x^{2} + 5)$
$\begin{matrix} x - 3 & = & 0 & or & x + 3 & = & 0 & or & x^{2} + 5 = 0 \\ x & = & 3 & or & x & = & - 3 & or & (no real solution) \end{matrix}$
The x-intercepts are $(3, 0)$ and $(–3, 0) .$
We can see these intercepts on the graph of the function shown in Figure 11. We can see that the function is even because $f (x) = f (- x) .$

Figure 11

Your turn

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

Given the polynomial function $f (x) = 2 x^{3} - 6 x^{2} - 20 x,$ determine the y- and x-intercepts.

How did it go?

PreviousDetermining the Intercepts of a Polynomial Function Next Determining the Number of Intercepts and Turning Points of a Polynomial

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 5.2, 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.