For our homework we had to read this one section which demonstrated the location principle (just in case this has a different name, I'll post it here:)

"If P(x) is a polynomial with real coefficients and a and b are real numbers such that P(a) and P(b) have opposite signs, then between a and b there is at least one real root r of the equation P(x)=0."

I definitely do NOT understand what the heck this means. Can someone please explain it, along with an example maybe? Any help is greatly appreciated, thanks!!

Do you know what a polynomial with real coefficients is?

It is a formula that is the sum of integer powers of x with constant real coefficients, plus a constant term. That is what they are calling P(x).

If there are two values a and b which, when substituted for x, result in P(a) and P(b) being of opposite signs, then there is at least one value of x between a and b for which P(x) = 0

Tammy mixes the letters S, C, H, O, O, and L thoroughly. Without looking, Allen draws one letter. Expressed as a fraction, decimal, and percentage, what is the probability that Allen will not select a consonant?

Tammy mixes the letters S, C, H, O, O, and L thoroughly. Without looking, Allen draws one letter. Expressed as a fraction, decimal, and percentage, what is the probability that Allen will not select a consonant?

Tammy mixes the letters S, C, H, O, O, and L thoroughly. Without looking, Allen draws one letter. Expressed as a fraction, decimal, and percentage, what is the probability that Allen will not select a consonant?

Of course, I'd be happy to explain the location principle and provide an example to help clarify things.

The location principle, also known as the Intermediate Value Theorem, states that if you have a polynomial function P(x) with real coefficients, and you have two real numbers a and b such that the function values P(a) and P(b) have opposite signs, then there must exist at least one real root of the equation P(x) = 0 between a and b.

To better understand this, let's break it down step by step:

1. Polynomial Function: A polynomial function is an equation that involves one or more variables raised to non-negative integer powers, summed together and multiplied by constants. For example, P(x) = 3x^2 - 2x + 1 is a polynomial function.

2. Real Coefficients: Real coefficients simply mean that all the coefficients of the polynomial function, such as the constants and the coefficients of the variables, are real numbers. Real numbers are numbers that can be expressed on the number line, including integers and decimals, but excluding numbers with an imaginary component.

3. Opposite Signs: When we say that P(a) and P(b) have opposite signs, it means that the function values at a and b result in different signs. For instance, if P(a) is positive and P(b) is negative, or vice versa, they have opposite signs.

So, according to the location principle, if you have a polynomial function with real coefficients and you can find two real numbers a and b such that P(a) and P(b) are of opposite signs, you can conclude that there exists at least one real root of the equation P(x) = 0 between a and b.

Now, let's go through an example to demonstrate this concept:

Example: Consider the polynomial function P(x) = x^3 - 4x^2 + x + 6.
Suppose we want to find a value of x between -2 and 0 such that P(x) = 0.

1. Calculate the function values at the endpoints:
P(-2) = (-2)^3 - 4(-2)^2 + (-2) + 6 = -8 - 16 - 2 + 6 = -20
P(0) = 0^3 - 4(0)^2 + 0 + 6 = 6

2. Determine if the function values have opposite signs:
P(-2) = -20 (negative) and P(0) = 6 (positive). Therefore, they have opposite signs.

According to the location principle, since P(-2) and P(0) have opposite signs, there must exist at least one real root of the equation P(x) = 0 between -2 and 0. In this case, when you graph the function, you will find that there is indeed a real root between these values.

I hope this explanation helps you understand the location principle better. If you have any further questions or need additional clarification, please feel free to ask!