If you are using the Factor Theorem and synthetic division to find real zeros of a given polynomial, how can Descartes' Rule of Signs increase the efficiency of the process?

furajat4h
2022-09-23
Answered

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Emaidedip6g

Answered 2022-09-24
Author has **11** answers

Descartes' Rule of Signs gives us a maximum of positive or negative real zeros of a function. If we know that, for example, some function f has 2 positive real zeros at most, and we have found those real zeros, we can ignore the other possible positive zeros and focus on the negative zeros.

For example: take the function

$$f(x)={x}^{3}-2{x}^{2}-5x+6$$

If f has any rational roots, they will be one of

$$\pm 1,\pm 2,\pm 3,\pm 6$$

Now, by Descartes' Rule of Signs, as f(x) has two sign changes, it will have 2 or 0 positive real zeros.However, we note that

$$f(-x)=-{x}^{3}-2{x}^{2}+5x+6$$

only has one sign change, and so it has exactly one negative real zero. Thus we can choose to start by the negative zeros, because once we have found one, we can move on to the positive. In this case, we find that x=-2 is a real zero, and so we can stop looking for negative zeros and look for the remaining positive zeros.

For example: take the function

$$f(x)={x}^{3}-2{x}^{2}-5x+6$$

If f has any rational roots, they will be one of

$$\pm 1,\pm 2,\pm 3,\pm 6$$

Now, by Descartes' Rule of Signs, as f(x) has two sign changes, it will have 2 or 0 positive real zeros.However, we note that

$$f(-x)=-{x}^{3}-2{x}^{2}+5x+6$$

only has one sign change, and so it has exactly one negative real zero. Thus we can choose to start by the negative zeros, because once we have found one, we can move on to the positive. In this case, we find that x=-2 is a real zero, and so we can stop looking for negative zeros and look for the remaining positive zeros.

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