Real Zero Bounds for Polynomial Functions

In summary, we can use the Intermediate Value Theorem and synthetic division to determine whether or not a given polynomial has a real zero between two numbers. For P(x) = x3 - 3x2 + 2x - 5, there is a real zero between 2 and 3. For P(x) = x4 + 2x3 + 2x2 - 5x + 3, there is no real zero between 0 and 1. And for P(x) = x3 - 2x2 - 5, there is no real zero between -1 and -2. However, for polynomials with odd degree, like P(x) = x3 - x2 + x
  • #1
Loonygirl
5
0
Use the Intermediate Value Theorem and synthetic division to determine whether or not the following polynomials have a real zero between the numbers given.

P(x) = x3 - 3x2 + 2x - 5; Is there a real zero between 2 and 3?

P(x) = x4 + 2x3 + 2x2 - 5x + 3; Is there a real zero between 0 and 1?

P(x) = x3 - 2x2 - 5; Is there a real zero between -1 and -2?

Find the upper and lower bounds for the following polynomials.

P(x) = x3 - x2 + x - 2

P(x) = x3 + 2x2 - 4
 
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  • #2
That looks very much like homework and I see no attempt on your part to do these problems!

(Odd degree polynomials do NOT have upper and lower bounds on the real numbers.)
 

What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a fundamental theorem in calculus that states that if a continuous function f(x) is defined on a closed interval [a,b] and takes on two values, say c and d, at the endpoints a and b respectively, then for any value between c and d (including c and d), there exists at least one point c within the interval [a,b] where f(c) is equal to that value.

How is the Intermediate Value Theorem used in mathematics?

The Intermediate Value Theorem is used to prove the existence of roots or solutions to equations, particularly in calculus and analysis. It is a key tool in proving the existence of solutions to many problems in mathematics, such as finding the roots of polynomials or solving differential equations.

What are the conditions for the Intermediate Value Theorem to hold?

For the Intermediate Value Theorem to hold, the function must be continuous on the closed interval [a,b]. This means that the function has no gaps, holes, or jumps within the interval. Additionally, the values c and d must lie between the endpoints a and b, and there must be a value within the interval where the function takes on the value between c and d.

Can the Intermediate Value Theorem be applied to any function?

No, the Intermediate Value Theorem only applies to continuous functions. If a function is not continuous, there may be gaps or jumps in the graph that prevent the theorem from being applied. Additionally, the function must be defined on a closed interval [a,b] for the theorem to hold.

What are the real-world applications of the Intermediate Value Theorem?

The Intermediate Value Theorem has many real-world applications in fields such as economics, physics, and engineering. For example, it can be used to prove the existence of solutions to equations in optimization problems or to determine the stability of dynamic systems in physics. It is also used in numerical methods for approximating solutions to equations and in computer graphics for rendering smooth images.

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