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Is it always true that between any two zeros of the derivative of any polynomial there is a zero of the polynomial?
Morl5001
No.
Consider the polynomial:
f(x) = x3 + 4x2 + 4x + 16
then
f'(x) = 3x2 + 8x + 4
= (3x + 2)(x + 2)
=> x = -2/3, -2 are the zeros of f'(x)
Using the second derivative:
f''(x) = 6x + 8
it can be seen that:
f''(-2) = -4 -> x = -2 is a maximum
f''(-2/3) = +4 -> x = -2/3 is a minimum
But plugging back into the original polynomial:
f(-2) = 16
f(-2/3) = 14 22/27
Between the zeros of the first derivative, the slope of the polynomial is negative so that the polynomial is always decreasing in value, but as the polynomial is greater than zero at the zeros of the first derivative, it cannot become zero between them.
That is it has no zeros between the zeros of its first derivative
f(x) = x3 + 4x2 + 4x + 16
= (x + 4)(x2 + 4)
has only 1 zero at x = -4.
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∙ 13y ago
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What is a quadratic polynomial which has no zeros?
A quadratic polynomial must have zeros, though they may be complex numbers.A quadratic polynomial with no real zeros is one whose discriminant b2-4ac is negative. Such a polynomial has no special name.
Is it always true that for any polynomial px if x is a zero of the derivative then x px is a maximum or minimum value of px?
No. The important decider is the second derivative of the polynomial (the gradient of the gradient of the polynomial) at the zero of the first derivative: If less than zero, then the point is a maximum If more than zero, then the point in a minimum If equal to zero, then the point is a point of inflection. Consider the polynomial f(x) = x3, then f'(x) = 3x2 f'(0) = 0 -> x = 0 could be a maximum, minimum or point of inflection. f''(x) = 6x f''(0) = 0 -> x = 0 is a point of inflection Points of inflection do not necessarily have a zero gradient, unlike maxima and minima which must. Points of inflection are the zeros of the second derivative of the polynomial.
What are the zeros of polynomials?
The values of the variables which make the polynomial equal to zero
Is it possible that the polynomial function doesn't have zeros?
In the real domain, yes. In the complex domain, no.
How do you graph a polynomial in order to solve for the Zeros?
Either graph the polynomial on graph paper manually or on a graphing calculator. If it is a "y=" polynomial, then the zeroes are the points or point where the polynomial touches the x-axis. If it is an "x=" polynomial, then the zeroes are the points or point where the polynomial touches the y-axis. If it touches neither, then it has no zeroes.
Is it always true that between any two zeros of any polynomial there is a zero of the derivative?
Yes.
Is it always true that the zeros of the derivative and the zeros of the polynomial always alternate in location along the horizontal axis?
A zero of the derivative will always appear between two zeroes of the polynomial. However, they do not always alternate. Sometimes two or more zeroes of the derivative will occur between two zeroes of a polynomial. This is often seen with quartic or quintic polynomials (polynomials with the highest exponent of 4th or 5th power).
What is a quadratic polynomial which has no zeros?
A quadratic polynomial must have zeros, though they may be complex numbers.A quadratic polynomial with no real zeros is one whose discriminant b2-4ac is negative. Such a polynomial has no special name.
Is it always true that for any polynomial px if x is a zero of the derivative then x px is a maximum or minimum value of px?
No. The important decider is the second derivative of the polynomial (the gradient of the gradient of the polynomial) at the zero of the first derivative: If less than zero, then the point is a maximum If more than zero, then the point in a minimum If equal to zero, then the point is a point of inflection. Consider the polynomial f(x) = x3, then f'(x) = 3x2 f'(0) = 0 -> x = 0 could be a maximum, minimum or point of inflection. f''(x) = 6x f''(0) = 0 -> x = 0 is a point of inflection Points of inflection do not necessarily have a zero gradient, unlike maxima and minima which must. Points of inflection are the zeros of the second derivative of the polynomial.
How do the zeros of a polynomial function help you determine the answer?
They tell you where the graph of the polynomial crosses the x-axis.Now, taking the derivative of the polynomial and setting that answer to zero tells you where the localized maximum and minimum values occur. Two values that have vast applications in almost any profession that uses statistics.
What do the zeros of a polynomial function represent on a graph?
The zeros of a polynomial represent the points at which the graph crosses (or touches) the x-axis.
-2,1,4?
Polynomial fuction in standard form with the given zeros
Sum and product of the zeros of a quadratic polynomial are -12 and -3 respectively what is the quadratic polynomial?
x2 + 15x +36
How do you find polynomial whose zeros are given?
when the equation is equal to zero. . .:)
What are the zeros of polynomials?
The values of the variables which make the polynomial equal to zero
Is the x-intercepts the same thing as zeros?
Yes, the places where the graph of a polynomial intercepts the x-axis are zeros. The value of y at those places must be 0 for the polynomial to intersect the x axis.
What is the factor theorem?
In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.The factor theorem states that a polynomial has a factor if and only if
