Question

What polynomial has roots of -4,-1, and 6?

Answer 1

a polynomial with the roots r1,r2 and r3 has factors of
(x-r1), (x-r2) and (x-r3) in it, it can be written in form
c(x-r1)ᵃ(x-r2)ᵇ(x-r3)ⁿ
where c is a constant and a,b, and n are the powers

so roots of -4,-1 and 6 are factors of
(x-(-4))(x-(-1))(x-6)=(x+4)(x+1)(x-6)
in expanded form
x³-x²-26x-24

Answer 2

The answer is x³-x²-26x-24

Explanation:

Firstly, any polynomial can be formed from its roots following the next rule:

Let r1, r2, r3,..., the roots of a polynomial P(x)

P(x)=C*(x-r1)*(x-r2)*(x-r3)*...

Where C is a real constant number excepting zero.

So for roots -4,-1 and 6, the polynomial is:

`C*(x-(-4))*(x-(-1))*(x-6)`

`C*(x+4)*(x+1)*(x-6)\\C*[(x^2+5*x+4)*(x-6)]\\C*[x^3+5*x^2+4*x-6*x^2-30*x-24]\\C*[x^3-x^2-26*x-24]\\`

Finally, considering C=1 (can be any real number excepting zero)

`x^3-x^2-26*x-24`