Question

Find a polynomial f(x) of degree 5 such that both of these properties hold:

f(x)-1 is divisible by (x-1)^3.

f(x) is divisible by x^3.

I have no idea what to do. Any help is appreciated. Will medal and fan!

Answer 1

Hi :)

Well, you sure know about polynomial remainders. The first property is that \(f(x)-1\) is divisible by \((x-1)^3\), it means that there is a polynomial of degree 2 such that:\[f(x)-1=(x-1)^3 g(x)\]or\[f(x)=1+(x-1)^3 g(x) \ \ \ \ (i)\]Second property is that \(f(x)\) is divisible \(x^3\), and this one leads to\[f(x)=x^3 h(x) \ \ \ (ii)\]where \(h(x)\) is a quadratic too.

Now from \((i)\) and \((ii)\) you get\[1+(x-1)^3 g(x)=x^3 h(x)\]and since \(g(x)\) and \(h(x)\) are some quadratics you can write\[1+(x-1)^3 (ax^2+bx+c)=x^3 (a'x^2+b'x+c')\]Now equate coefficients of both sides and evaluate \(a, b, c, a', b'\) and \(c'\).

Answer 2

* it means that there is a polynomial of degree 2 like \(g(x)\) such that: