1. ## Polynomials and remainders

Prove that when a polynomial $f(x)$ is divided by $ax + b$, where a is not equal to 0 the remainder is $f(-b/a)$.
Find the polynomial in x of the third degree, which vanishes when $x =-1$ and $x=2$, has the value 8 when $x=0$ and leaves the remainder $16/3$ when divided by $3x +2$. someone please help me to figure this problem out. Thanks in advance.

2. Originally Posted by scrible
I have a log problem bothering me the question goes Prove that when a polynomial $f(x)$ is divided by $ax + b$, where a is not equal to 0 the remainder is $f(-b/a)$.
Find the polynomial in x of the third degree, which vanishes when $x =-1$ and $x=2$, has the value 8 when $x=0$ and leaves the remainder $16/3$ when divided by $3x +2$. someone please help me to figure this problem out. Thanks in advance.
Notice that a(-b/a)+b=0. If we divide f(x) by ax+b then there is some g(x) such that f(x)=g(x)(ax+b)+r(x), where r is the remainder. g(x) has a degree one less than f(x), so if f(x)=h*x^5+.... then g(x)=u*x^4_....

f(x)=g(x)(ax+b)+r
f(-b/a)=g(-b/a)(a(-b/a+b))+r
f(-b/a)=g(-b/a)(0)+r
f(-b/a)=r

3. What do these have to do with logarithms?