Answer:
[tex]P(x)=x^3-5x^2+2x+8[/tex]
Step-by-step explanation:
If the polynomial has the following zeros: x= 4, x=2, and x=-1, then it must have the following factors which guarantee the polynomial will render zero at the given points:
[tex]P(x)=C (x-4)\, (x-2) \,(x+1)[/tex]
where C is any multiplicative constant. Since there is no other condition imposed on the polynomial, we can adopt a simple constant C = 1 to facilitate our final calculation of writing the polynomial in standard form:
[tex]P(x)=(x-4)\, (x-2) \,(x+1)\\P(x)=(x^2-2x-4x+8)\,(x+1)\\P(x)= (x^2-6x+8)\,(x+1)\\P(x)=x^3+x^2-6x^2-6x+8x+8\\P(x)=x^3-5x^2+2x+8[/tex]
