# polynomials

• Sep 4th 2010, 12:35 AM
earthboy
polynomials
1. a polynomial f(x) leaves remainder 15 and (2x+1) when divided by (x-3) and (x-1)^2 respectively.then,what is the remainder when it is divided by (x-3){(x-1)^2}

2. find all polynomials P satisfying p(x+1)=p(x)+2x+1

3. if all the roots of the polynomial equation
x^4-4x^3+ax^2+bx+1=0 are positive real numbers,show that all the roots are equal.

question 2:
i have observed that p(x)=x^2 satisfies the given condition.do we now have to substitute p(x) in the condition ?
how should we substitute?
• Sep 4th 2010, 09:45 PM
CaptainBlack
Quote:

Originally Posted by earthboy

2. find all polynomials P satisfying p(x+1)=p(x)+2x+1

Suppose:

\$\displaystyle p(x)=a_nx^n+a_{n-1}x^{n-1}+..._a_1x+a_0\$

with \$\displaystyle a_n \ne 0\$, is such a polynomial. Then:

\$\displaystyle p(x+1)=a_n(x+1)^n+a_{n-1}(x+1)^{n-1}+..._a_1(x+1)+a_0\$

...,,.... \$\displaystyle =a_nx^n + (na_n+a_{n-1})x^{n-1}+ ... \$

which implies either that \$\displaystyle (na_n+a_{n-1})=a_{n-1}\$ (which is not possible as \$\displaystyle a_n \ne 0\$ ) or that \$\displaystyle n=2\$ and \$\displaystyle a_2=1\$

etc

CB