Option A:
[tex]g(x) =x^4+x^3-7x^2-x+6[/tex]
Solution:
Given data: Zeroes are 1, 2, –3, and –1.
To find the polynomial of the function for the given zeroes.
If 1 is a root of the polynomial then the factor is (x – 1).
If 2 is a root of the polynomial then the factor is (x – 2).
If –3 is a root of the polynomial then the factor is (x – (–3)) = (x + 3).
If –1 is a root of the polynomial then the factor is (x – (–1)) = (x + 1).
On multiplying the factors, we get the polynomial of the function.
[tex]\Rightarrow\ g(x)=(x-1)(x-2)(x+3)(x+1)[/tex]
[tex]\Rightarrow\ \ \ \ \ \ \ \ =(x^2-2x-x+2)(x^2+x+3x+3)[/tex]
[tex]\Rightarrow\ \ \ \ \ \ \ \ =(x^2-3x+2)(x^2+4x+3)[/tex]
Now multiplying each term of the first factor by each term of the second.
[tex]\Rightarrow\ \ \ \ \ \ \ \ =x^2(x^2+4x+3)-3x(x^2+4x+3)+2(x^2+4x+3)[/tex]
[tex]\Rightarrow\ \ \ \ \ \ \ \ =(x^4+4x^3+3x^2)+(-3x^3-12x^2-9x)+(2x^2+8x+6)[/tex]
Removing brackets in each term.
[tex]\Rightarrow\ \ \ \ \ \ \ \ =x^4+4x^3+3x^2-3x^3-12x^2-9x+2x^2+8x+6[/tex]
Combine the like terms and simplifying.
[tex]\Rightarrow\ \ \ \ \ \ \ \ =x^4+4x^3-3x^3+3x^2-12x^2+2x^2-9x+8x+6[/tex]
[tex]\Rightarrow\ \ \ \ \ \ \ \ =x^4+x^3-7x^2-x+6[/tex]
[tex]\Rightarrow \ g(x) =x^4+x^3-7x^2-x+6[/tex]
Option A is the correct answer.
Hence [tex]g(x) =x^4+x^3-7x^2-x+6[/tex].
