To determine the derivative of a polynomial, you can use the power rule or the product rule. The power rule states that the derivative of xn is n*x^(n-1). The product rule states that the derivative of f(x)*g(x) is f'(x)*g(x) + f(x)*g'(x), where f'(x) and g'(x) are the derivatives of f(x) and g(x) respectively.
A polynomial has a zero in an interval if there exists a value of x within that interval that makes the polynomial equal to zero. This means that the graph of the polynomial will intersect the x-axis at least once within that interval.
To show that a polynomial has a zero in an interval, you can use the intermediate value theorem. This theorem states that if a continuous function f(x) has values of a and b at two different points in an interval, then f(x) must take on every value between a and b at some point within that interval.
Yes, a polynomial can have more than one zero in an interval. This can happen when the graph of the polynomial crosses the x-axis multiple times within that interval.
If the derivative of a polynomial is positive at one point in an interval and negative at another point in the same interval, then by the intermediate value theorem, the polynomial must have a zero within that interval. This is because the derivative tells us the slope of the polynomial at any given point, and a change in sign of the derivative indicates a change in direction of the polynomial's graph, which must cross the x-axis at some point.