Math Is Fun Forum

For each of the following, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain:

e) f(x) = |x| * \sqrt{x}

Thanks!

1) Let f(x) = x^3+bx+c, where b and c are integers.

If f(5+\sqrt 3)=0, determine b+c.

I know that 5-\sqrt 3 is also a root, but how to continue?
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2) Find a monic quartic polynomial f(x) with rational coefficients whose roots include

. Give your answer in expanded form.

???
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3) Suppose $P(x)$ is a polynomial of smallest possible degree such that:

-P(x) has rational coefficients

-P(-3) = P(\sqrt 7) = P(1-\sqrt 6) = 0

-P(-1) = 8

Determine the value of P(0).

???

Part 1:

Let f(x) and g(x) be polynomials.

Suppose f(x)=0 for exactly three values of x: namely, x=-3,4, and 8.

Suppose g(x)=0 for exactly five values of x: namely, x=-5,-3,2,4, and 8.

Is it necessarily true that g(x) is divisible by f(x)? If so, carefully explain why. If not, give an example where g(x) is not divisible by f(x).

Part 2:

Generalize: for arbitrary polynomials f(x) and g(x), what do we need to know about the zeroes (including complex zeroes) of f(x) and g(x) to infer that g(x) is divisible by f(x)?

(If your answer to Part 1 was "yes", then stating the generalization should be straightforward. If your answer to Part 1 was "no", then try to salvage the idea by imposing extra conditions as needed. Either way, prove your generalization.)

How to do this problem?

1) The polynomial

is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$.

     
     Do I plug in x=4? What would I do???
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2) Suppose the polynomial

is of degree

and satisfies

,

,

, and

.

Determine the value of

.

     
     How to do this?
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1) Let

,

, and

be points on a circle. Let

and the tangent to the circle at

intersect at

. If

,

, and

, then find

.

2) In triangle

,

,

, and

. Let

be the point on the circumcircle of triangle

so that

. Find

.

Please help!

We are given a regular heptagon of side length 1. Let S be the set of points that are within a distance of 1 from some point on or inside the heptagon, but not including the heptagon itself. Find the area of S.

Please help!!!

Can you guys please help me with these problems?

1. In the triangle shown, n is a positive integer, and angle A > angle B > angle C. How many possible values of n are there?
Diagram