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Qwerty90
 3 years ago
Question about how to solve: Determine the zeros of f(x) = x4 – x3 + 7x2 – 9x – 18.
I am solving this problem using all of the polynomial ways such as decartes rule, fundamental therom and rational root. But it seems when I solve it that way, I either come up one answer short or a completely different answer from what I see everyone else get. How would you solve this and what is the answer?
Qwerty90
 3 years ago
Question about how to solve: Determine the zeros of f(x) = x4 – x3 + 7x2 – 9x – 18. I am solving this problem using all of the polynomial ways such as decartes rule, fundamental therom and rational root. But it seems when I solve it that way, I either come up one answer short or a completely different answer from what I see everyone else get. How would you solve this and what is the answer?

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ZeHanz
 3 years ago
Best ResponseYou've already chosen the best response.0RRT says: Rational solutions can only be factors of 18 divided by factors of the coefficient of x^4. Rational roots therefore have the form:\[\pm 18, \pm9, \pm6, \pm3, \pm2, \pm1\]If you try (begin with the simplest, of course, so 1 or 1), you get 1 as a root. You can now write f(x) as (x+1)(x³ ..........). To get the 3rd degree part, you can do a long division or a synthetic division. As soon as you have found it, you can do this trick again: I think 2 is a root as well (just try it), so you can factor out x2 to get (x+1)(x2)(x²........). Now you can esily check if there are more rational or even real roots.

ZeHanz
 3 years ago
Best ResponseYou've already chosen the best response.0Could it be that you are not familiar with synthetic division? It's a real handy trick!

Qwerty90
 3 years ago
Best ResponseYou've already chosen the best response.0No I know how to do synthetic division, its just after that where everything get messy.

ZeHanz
 3 years ago
Best ResponseYou've already chosen the best response.0OK, so you've found: f(x)=(x+1)(x2)(x²+9). As you can see, the last factor x²+9 has no real roots, only complex ones: 3i and 3i.

Qwerty90
 3 years ago
Best ResponseYou've already chosen the best response.0So, basically after you found which zeros is part of the function you list it in parentheses as the conjugate. Now my last question is where did you get the x+9 from? i know that 9 is the constant but what about the x^2?

ZeHanz
 3 years ago
Best ResponseYou've already chosen the best response.0After the first synthetic division, I got (x+1)(x³2x²+9x18). Then the number 2 is also a zero, so I divided x³2x²+9x18 by x2. That gave me x²+9
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