Find the number of polynomials f(x) that satisfy all of the
following conditions:
f(x) is a monic polynomial,
f(x) has degree 1000,
f(x) has integer coefficients,
f(x) divides
f(2x^3+x)
http://www.google.com/search?q=Find+the+number+of+polynomials+f(x)+that+satisfy+all+of+the+following+conditions%3A&oq=Find+the+number+of+polynomials+f(x)+that+satisfy+all+of+the+following+conditions%3A&aqs=chrome.0.57j62.4880j0&sourceid=chrome&ie=UTF-8#sclient=psy-ab&q=brilliant+Find+the+number+of+polynomials+f(x)+that+satisfy+all+of+the+following+conditions:&oq=brilliant+Find+the+number+of+polynomials+f(x)+that+satisfy+all+of+the+following+conditions:&gs_l=serp.3...11592.13491.0.13851.10.10.0.0.0.0.220.1571.0j9j1.10.0...0.0.0..1c.1.17.psy-ab.YdKvWIF7Svo&pbx=1&bav=on.2,or.r_cp.r_qf.&bvm=bv.48705608,d.eWU&fp=6ca84d98d48f74bf&biw=1440&bih=706
Step 1; Factor f(x) = (x - r_1)...(x - r_1000)
Step 2: Hint hint: something about roots and circle centered at r_i of radius 2 |r_i|^3
Step 3: triangle inequality --> implication: same argument.
Step 4: Verify
This is a brilliant question, therefore i will not give full answer. i hope the hints are enough to set one on the right track.
To find the number of polynomials that satisfy the given conditions, we need to break the problem down into smaller steps. Here's how you can approach it:
Step 1: Determine the form of f(x)
Since f(x) is a monic polynomial with degree 1000 and integer coefficients, we can represent it as:
f(x) = x^1000 + a_999*x^999 + a_998*x^998 + ... + a_1*x + a_0
Step 2: Evaluate f(2x^3 + x)
Substitute 2x^3 + x for x in f(x) and simplify:
f(2x^3 + x) = (2x^3 + x)^1000 + a_999*(2x^3 + x)^999 + a_998*(2x^3 + x)^998 + ... + a_1*(2x^3 + x) + a_0
Step 3: Simplify f(2x^3 + x)
Expand the binomial terms using the binomial theorem. This involves raising each term in (2x^3 + x) to the power of 1000, 999, 998, etc., and multiplying them by the corresponding coefficients. The resulting expression will be a polynomial in terms of x.
Step 4: Express f(2x^3 + x) as a product of f(x) and a quotient polynomial
We can write f(2x^3 + x) as a product of f(x) and a quotient polynomial if f(x) divides f(2x^3 + x) without a remainder. This can be expressed as:
f(2x^3 + x) = f(x) * q(x)
Step 5: Compare the coefficients
Compare the coefficients of corresponding powers of x on both sides of the equation f(2x^3 + x) = f(x) * q(x). Since both f(x) and q(x) have integer coefficients, we need to find the number of solutions f(x) can have for each coefficient.
Step 6: Determine the number of polynomials
Count the number of possible integer values for each coefficient.
Note: Calculating the actual values of the coefficients may be computationally intensive, so it might be more practical to focus on determining the number of possible values rather than actually finding the specific polynomials.
By following these steps, you can determine the number of polynomials f(x) that satisfy all the given conditions.