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Use Bolzano’s theorem to isolate real roots of given polynomials

A real value is a root of a function if . We say we have isolated a real root if we find an interval such that and no other real roots are in the interval. Use Bolzano’s Theorem to isolate the real roots of the following:

1. .
2. .
3. .

1. We have

2. We have

3. We have

Prove a polynomial with opposite signed first and last coefficients has at least one positive zero

Define a polynomial

such that and have opposite signs. Prove there is some such that .

Proof. Since

we see that has the same sign as .
Next,

Then we claim that for sufficiently large ,

hence, will have the same sign as for sufficiently large , since and so

(So, when we multiply a positive number by the result will have the same sign as .)

So, we need to show that the claimed term is indeed positive. First,

This is true since for each term in the sum

Now, since we are showing there is sufficiently large such that our claim is true, we let

Since , we know for all , so

This proves our claim, and so has the same sign as for sufficiently large . Hence, and have different signs (since and had different signs by assumption); thus, there is some such that

Compute the integral

Compute the following integral:

We have

Compute the integral

Compute the following integral:

We have

Compute the integral

Compute the following integral:

We have

Compute the integral

Compute the following integral:

We have

Compute the integral

Compute the following integral:

We have

Compute the integral

Compute the following integral:

We have

Compute the integral

Compute the following integral:

We have

Compute the integral

Compute the following integral:

We have