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Exercises Polynomials and factoring

Example 2 from Algebra and Trigonometry, 5.3 Graphs of Polynomial Functions

Finding the x-Intercepts of a Polynomial Function by Factoring

Find the x-intercepts of $f (x) = x^{6} - 3 x^{4} + 2 x^{2} .$

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.

Solution
We can attempt to factor this polynomial to find solutions for $f (x) = 0.$

$\begin{matrix} x^{6} - 3 x^{4} + 2 x^{2} & = & 0 & \begin{matrix} Factor out the greatest \\ common factor . \end{matrix} \\ x^{2} (x^{4} - 3 x^{2} + 2) & = & 0 & Factor the trinomial . \\ x^{2} (x^{2} - 1) (x^{2} - 2) & = & 0 & Set each factor equal to zero . \end{matrix}$
$\begin{matrix} (x^{2} - 1) & = & 0 & (x^{2} - 2) & = & 0 \\ x^{2} & = & 0 & or & x^{2} & = & 1 & or & x^{2} & = & 2 \\ x & = & 0 & x & = & ±1 & x & = & \pm \sqrt{2} \end{matrix}$
This gives us five x-intercepts: $(0, 0), (1, 0), (−1, 0), (\sqrt{2}, 0),$ and $(- \sqrt{2}, 0) .$ See Figure 3. We can see that this is an even function because it is symmetric about the y-axis.

Figure 3

How did it go?

PreviousRecognizing Polynomial Functions Next Finding the x-Intercepts of a Polynomial Function by Factoring

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.