A positive zero of a polynomial is a value of x that makes the polynomial equal to zero when plugged in for x. In other words, it is a value that satisfies the equation P(x) = 0 and is greater than 0.
To prove that a polynomial has a positive zero, we can use the Intermediate Value Theorem. This theorem states that if a continuous function has values of opposite signs at two points, then there exists at least one root between those two points. In the case of the polynomial P(x), we can show that P(0) = -1 and P(1) = 1, which means there must be at least one positive zero between 0 and 1.
Proving that a polynomial has a positive zero is important because it allows us to find the roots of the polynomial, which can provide valuable information about the behavior and characteristics of the polynomial. Additionally, it can help us solve equations and make predictions in various fields such as physics, engineering, and economics.
The significance of the polynomial P(x)=x^{13}+x^7-x-1 having 1 positive zero is that it tells us that there is at least one value of x that makes the polynomial equal to zero. This means that the polynomial has at least one real root, which can be useful in solving equations and understanding the behavior of the polynomial.
Yes, a polynomial can have more than one positive zero. In fact, the polynomial P(x)=x^{13}+x^7-x-1 has exactly one positive zero, but it also has other roots that are not positive. This is because polynomials can have complex roots, which are not real numbers. However, the converse is not true - a polynomial cannot have more positive zeros than its degree.