Zeros of a Polynomial Function Lesson

Over the course of the last few lessons, we have been discussing various tools and techniques that can be used to find the zeros for a polynomial function. Let's walk through an example of how to find the zeros for a polynomial function.
Example #1: Find all zeros. $$f(x)=x^3 - 3x^2 - 12x + 10$$ First, check to see if you can factor the polynomial as it stands. In this case, we can't factor using grouping. Let's move on and think about a few things.

• From the fundamental theorem of algebra, we know that we have at most 3 distinct solutions
• From the rational roots test, we obtain a list of possible rational roots:
  • $$\pm (1, 2, 5, 10)$$
• From Descartes' rule of signs, we obtain a list of possible positive and negative real roots:
  • + roots: 2 or 0
  • - roots: 1

From here, we can use synthetic division, along with our upper and lower bound rules to narrow down the possibilities. Eventually, we will find that 5 is a zero: $$\frac{x^3-3x^2-12x+10}{x-5}=x^2 + 2x - 2$$ We can use this to factor our polynomial: $$f(x)=(x - 5)(x^2 + 2x - 2)$$ We can take the quadratic factor and use our quadratic formula to obtain: $$x=-1 \pm \sqrt{3}$$ This gives us the following zeros for our polynomial function: $$x=-1 \pm \sqrt{3}, 5$$

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