I'm learning Precalculus right now and I have a question about the upper and lowers bounds of a polynomial. My textbook defines the lower bound (a) and upper bound (b) as a ≤ c ≤ b, where every real zero of the polynomial satisfies c. This makes sense because graphically on the x-axis, the lower bound would be to the left of all the zeros and the upper bound to the right.
So the way to find them is to use synthetic division. To find the upper bound, the row that contains the quotient and remainder would have no negative numbers. To find the lower lower bound, the row with the quotient and remainder would have alternating nonpositive and nonnegative numbers.
This all makes sense to me. However, how exactly do you find them; is it simply by trial and error? Also, it is correct that an infinite amount of numbers can satisfy the conditions to be upper and lower bounds? For instance, the polynomial P(x) = (x - 1)(x + 2) has roots 1 and -2. Does this mean the upper bound could be anything greater than or equal to 1, and the lower bound could be anything less than or equal to -2. Basically, I'm asking what integers would you use because there seems to be an infinite number of integers that satisfy a and b in a ≤ c ≤ b for a given polynomial. I hope my question makes, and any help will be greatly appreciated.
So the way to find them is to use synthetic division. To find the upper bound, the row that contains the quotient and remainder would have no negative numbers. To find the lower lower bound, the row with the quotient and remainder would have alternating nonpositive and nonnegative numbers.
This all makes sense to me. However, how exactly do you find them; is it simply by trial and error? Also, it is correct that an infinite amount of numbers can satisfy the conditions to be upper and lower bounds? For instance, the polynomial P(x) = (x - 1)(x + 2) has roots 1 and -2. Does this mean the upper bound could be anything greater than or equal to 1, and the lower bound could be anything less than or equal to -2. Basically, I'm asking what integers would you use because there seems to be an infinite number of integers that satisfy a and b in a ≤ c ≤ b for a given polynomial. I hope my question makes, and any help will be greatly appreciated.
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Typically, we define the lower bound as the greatest integer that is less than or equal to all of the polynomial's zeroes and the upper bound as the smallest integer that is equal to or greater than all of the polynomial's zeroes. In the case of P(x) = (x - 1)(x + 2), the lower bound will be -2 and the upper bound is 1.
Without a graph, trial-and-error is your only option. Most problems that you will encounter will not have ridiculous lower or upper bounds; the two bounds should be relatively easy to find by trial-and-error. If you have a graph of the function, you can easily obtain the lower and upper bounds.
I hope this helps!
Without a graph, trial-and-error is your only option. Most problems that you will encounter will not have ridiculous lower or upper bounds; the two bounds should be relatively easy to find by trial-and-error. If you have a graph of the function, you can easily obtain the lower and upper bounds.
I hope this helps!
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The 'bounds' for a polynomial function aren't really something we solve for because they're usually obvious. You know that every polynomial with an even degree will either go to plus or minus infinity, and have an upper or lower bound at its largest vertex. You know that every polynomial with an odd degree will not be bounded, and go to both plus and minus infinity. It really helps to look at some graphs of different functions. You should notice a pattern with the bounds if you graph y = x, then y = x^2, then y = x^3 and so on.
If you know that the bounds are equal to roots then you would have to find the roots.
If you know that the bounds are equal to roots then you would have to find the roots.