How do I write a polynomial function degree with integral coefficients that has given zeros, when your given -5 mult. 3

Question

I need help understanding this problem, it's for my math review and I have no idea how to do this.

Answer 1

The general process is if the polynomial with rational coefficients has roots are r₁ , r₂ ,... , r_n with corresponding multiplicities m₁ , m₂ , ... , m_n , then it equals

 

p(x) = a(x - r₁)^m_1(x - r₂)^m_2...(x - r_n)^m_n

 

"a" can be anything.

 

In your case, the only root is r = -5 with corresponding multiplicity m = 3. So if we let a=1 for simplicity, we get

 

p(x) = (x + 5)³.

 

We can expand this and get

 

p(x) = x³ + 15x² + 75x + 125

 

A technical note. The factorization used is over the splitting field, which is defined as the smallest field of numbers containing both the coefficients together with the polynomial's roots.