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Why any even degree polynomial will have at least a maxima or a minima?
Wiki User
∙ 15y ago
Because when the variable becomes sufficiently large, positive or negative, it's magnitude is much greater than any of the coefficients. Thus, the term with the highest power easily dominates with the other terms, which then contribute an insignificant amount.
For example, take y=x^4-985x^3-17. When x=±1000000000000000000, the first term is this large number multiplied four times, whereas the second term only has it multiplied three times, with a comparatively much smaller value and the 17 is even less significant.
A nonzero real number raised to an even power is always positive, so the sign at both extremes will be that of the coefficient; thus, both extremes will either be positive or negative.
When the variable is near 0, the value becomes near that of the constant, which is undeniable smaller that whatever astronomically high value the variable might attain. So, the graph must go from some extremely high value to some relatively small value, and then back to another extremely high value of the same sign (positive or negative) that it started out.
Wiki User
∙ 15y ago
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What is the least degree a polynomial could have with an imaginary root with a multiplicity of three?
Since the question did not specify a rational polynomial, the answer is a polynomial of degree 3.
What is the coefficient of the term of degree 1 in the polynomial?
There's no way for me to tell until you show methe polynomial, or at least the term of degree 1 .
Is the degree of a polynomial the greatest or least value of the variable's exponent?
The greatest.
What is the least degree of a polynomial with the roots 3 0 -3 and 1?
The polynomial P(x)=(x-3)(x-0)(x+3)(x-1) is of the fourth degree.
The polynomial given below has rootss?
You forgot to copy the polynomial. However, the Fundamental Theorem of Algebra states that every polynomial has at least one root, if complex roots are allowed. If a polynomial has only real coefficients, and it it of odd degree, it will also have at least one real solution.
If you are asked to write a polynomial function of least degree with real coefficients and with zeros of 2 and i square roots of seven what would be the degree of the polynomial also wright equation?
3y2-5xyz yay i figured it out!!!!
Is it true that the degree of polynomial function determine the number of real roots?
Sort of... but not entirely. Assuming the polynomial's coefficients are real, the polynomial either has as many real roots as its degree, or an even number less. Thus, a polynomial of degree 4 can have 4, 2, or 0 real roots; while a polynomial of degree 5 has either 5, 3, or 1 real roots. So, polynomial of odd degree (with real coefficients) will always have at least one real root. For a polynomial of even degree, this is not guaranteed. (In case you are interested about the reason for the rule stated above: this is related to the fact that any complex roots in such a polynomial occur in conjugate pairs; for example: if 5 + 2i is a root, then 5 - 2i is also a root.)
Do all polynomials have at least one minimum?
Polynomials of an even degree will always have either a minimum point, or a maximum point, or both.Polynomials of an odd degree may or may not have minima or maxima. If, for example, a polynomial function is simply a transformation of xn, there will be no turning points. For example:f(x) = x5 + 5x4 + 10x3 + 10x2 + 5x + 1 = (x+1)5f'(x) = 5(x+1)4There is only one solution for f'(x) = 0, which is of course x = -1. Since the range of f(x) includes all the real numbers, it follows that this solution represents a point of inflection, and not a turning point.If a polynomial of odd degree does have any turning points, it will have at least one minimum point. It cannot have maximum points only.* * * * *Polynomials of an odd degree cannot have a global maximum or minimum because if the leading coefficient is positive, it goes asymptotically from minus infinity to plus infinity and the other way around if the leading coefficient is negative.
What are the conditions for error to be minimum in least squares approximation?
Let's start with a first degree polynomial equation:This is a line with slope a. We know that a line will connect any two points. So, a first degree polynomial equation is an exact fit through any two points with distinct x coordinates.If we increase the order of the equation to a second degree polynomial, we get:This will exactly fit a simple curve to three points.If we increase the order of the equation to a third degree polynomial, we get:This will exactly fit four points.if we have more than n + 1 constraints (n being the degree of the polynomial), we can still run the polynomial curve through those constraints. An exact fit to all constraints is not certain (but might happen, for example, in the case of a first degree polynomial exactly fitting three collinear points). In general, however, some method is then needed to evaluate each approximation. The least squares method is one way to compare the deviations.
How do you write polynomial of least degree with roots 6 and -5?
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If f is continuous for all x and y and has two relative minima then f must have at least one relative maximum?
Yes.If you find 2 relative minima and the function is continuous, there must be exactly one point between these minima with the highest value in that interval. This point is a relative maxima.Think of temperature for example (it is continuous).
Quadratic equation definition?
An equation of the second degree, meaning it contains at least one term that is squared.
