Reconstruct Quartics from roots

Question

Not sure how to reconstruct an equation from roots. The roots are $-1/2, 1/2, 2, 3$ and the equation is $4x^4-20x^3+23x^2+5x-6.$

Answer 1

Let us say an equation has roots ($a_1,a_2,...a_k$)

Then, the polynomial must be of the form $(x-a_1)(x-a_2)...(x-a_k)$

Simply multiply and expand to get $(x-\frac{1}{2})(x+\frac{1}{2})(x-2)(x-3)=0$

By multiplying both sides by 4 we get $(2x-1)(2x+1)(x-2)(x-3)=0$ which expands to give $4x^4−20x^3+23x^2+5x−6= 0$