Answer: The polynomial has a total of 5 zeroes, can have a maximum of 5 positive real roots and a maximum of 4 negative real roots.
Step-by-step explanation: The given polynomial is
[tex]P(x)=x^5-9x^4+13x^3+57x^2-120.[/tex]
We are to find the total number of zeroes, number of maximum positive real roots and number of maximum negative real roots of the above polynomial.
Since the degree of the polynomial is 5, so there will be 5 zeroes of the polynomial P(x).
Let, 'a', 'b', 'c', ''d' and 'e' be the zeroes of P(x).
Then, according to the relations between roots and coefficients, we have
[tex]a\times b\times c\times d\times e=-(-120)\\\\\Rightarrow abcde=120.[/tex]
So, the product of the zeroes is positive. This is possible if
(i) the number of positive zeroes are 1, negative zeroes are 4.
(ii) the number of positive zeroes are 3, negative zeroes are 2.
(iii) the number of positive zeroes are 5, negative zeroes are 0.
Therefore, the maximum number of positive zeroes is 5 and the maximum number of negative zeroes is 4.
Thus, the polynomial has a total of 5 zeroes, can have a maximum of 5 positive real roots and a maximum of 4 negative real roots.
