[CSAT][CAT] How to find maximum positive and negative roots of a polynomial
To find the maximum number of positive and negative real roots of a polynomial f(x), we should consider Descartes’ rule of signs. This rules does not tell the exact number of positive or negative roots.
Descartes’ Sign Rule : The rule says that in a polynomial f(x), the maximum number of positive real roots can be ascertained by counting the number of sign changes in its coefficients.
The actual number of positive real roots maybe the maximum number or a number reduced by a multiple of 2.
For example in f(x) = 5x^6+3x^5-2x^4+6x^3+2x^2 – 7x +8, there are 4 sign changes +, – , + , – , +
So f(x) can have a maximum of 4 positive real roots or 2 or 0.
Similarly for negative real roots, we count the sign changes in f(-x)
Here f(-x)= 5x^6 – 3x^5 – 2x^4– 6x^3+ 2x^2 +7x +8. There are 2 sign changes +, -, +
Hence it can have a maximum number of 2 negative real roots or 0.
The exclusion of multiples of 2 is because the polynomial may have complex roots which always come in pairs.
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