My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
October 31st, 2015, 06:19 AM   #1
Newbie
 
Joined: Oct 2015
From: Edinburgh

Posts: 6
Thanks: 0

Post Polynomial problem

Can you help me with this problem please:
Find all pairs of polynomials A, B with real coefficients that for each real number expressions
A(x^2 + 1) = B(x)^2 + 2x
B(x^2 + 1) = A(x)^2
are true.
MrPres is offline  
 
November 2nd, 2015, 07:38 AM   #2
Newbie
 
Joined: Oct 2015
From: Edinburgh

Posts: 6
Thanks: 0

Any ideas?
MrPres is offline  
November 2nd, 2015, 09:25 AM   #3
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
The second line implies that the degree of A is the same as the degree of B.

B(x^2 + 1) has only terms with even powers, so the powers of terms in A(x) with nonzero coefficients must either be all even or all odd.

A(x^2 + 1) has only terms with even powers, so the powers of terms in B(x)^2 + 2x with nonzero coefficients must either be all even or all odd. Let B(x) = a + bx + O(x^2), then B(x)^2 = a^2 + 2abx + O(x^2) and so B(x)^2 + 2x = a^2 + (2ab + 2)x + O(x^2) and so either a or 2ab + 2 (or both) must be 0. If a is nonzero, that is, if B(x) is not a multiple of x, then b = -1/a.
CRGreathouse is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
polynomial, polynomials, problem



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
POLYNOMIAL PROBLEM!!!-3(x^2+10x+25)-4(x+5) taytay22 Pre-Calculus 2 August 22nd, 2015 12:13 PM
Polynomial Problem f(x^2+2)=x^4+10x^2+4... DapperDub Pre-Calculus 3 August 21st, 2015 07:48 AM
Polynomial problem KyVanchhay Algebra 2 June 27th, 2013 01:07 PM
problem with polynomial nukem4111 Algebra 0 April 13th, 2013 11:34 AM
A polynomial problem Pell's fish Number Theory 5 August 30th, 2010 03:04 PM





Copyright © 2019 My Math Forum. All rights reserved.