Find the polynomial

Question

Find a degree 3 polynomial having zeros -1,1 and 6 and the coefficient of x³ equal 1.

Answer 1

We start with the factored form of a polynomial with roots x = -1, x = 1, and x = 6. That is, we have

(x+1)(x-1)(x-6).

However, we can see that this expands easily so that we have

(x+1)(x-1)(x-6)=(x²-1)(x-6)=x³-6x²-x+6.

Since the coefficient of x³ is 1, we have that this is the correct order-three polynomial (or, rather, cubic).

Notice that if the polynomial if the equation had required a coefficient of x³ different from 1, we could have simply scaled the whole polynomial by that value since this would not change the roots of the polynomial.

Hope that helps!