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:)

Question

"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!!

Answer 1

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

Answer 2

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?