The possible formula for a fourth degree polynomial g is:
[tex]g(x)=\dfrac{1}{4}(x^4-3x^3-12x^2+20x+48)[/tex]
We know that if a polynomial has zeros as a,b,c and d then the possible polynomial form is given by:
[tex]f(x)=m(x-a)(x-b)(x-c)(x-d)[/tex]
Here the polynomial g has a double zero at -2, g(4) = 0, g(3) = 0.
This means that the polynomial g(x) is given by:
[tex]g(x)=m(x-(-2))^2(x-4)(x-3)\\\\i.e.\\\\g(x)=m(x+2)^2(x-4)(x-3)\\\\i.e.\\\\g(x)=m(x^2+2^2+2\times 2\times x)(x(x-3)-4(x-3))\\\\i.e.\\\\g(x)=m(x^2+4+4x)(x^2-3x-4x+12)\\\\i.e.\\\\g(x)=m(x^2+4+4x)(x^2-7x+12)\\\\i.e.\\\\g(x)=m[x^2(x^2-7x+12)+4(x^2-7x+12)+4x(x^2-7x+12)]\\\\i.e.\\\\g(x)=m[x^4-7x^3+12x^2+4x^2-28x+48+4x^3-28x^2+48x]\\\\i.e.\\\\g(x)=m[x^4-7x^3+4x^3+12x^2+4x^2-28x^2-28x+48x+48]\\\\i.e.\\\\g(x)=m[x^4-3x^3-12x^2+20x+48][/tex]
Also,
[tex]g(0)=12[/tex]
i.e.
[tex]48m=12\\\\i.e.\\\\m=\dfrac{12}{48}\\\\i.e.\\\\m=\dfrac{1}{4}[/tex]
Hence, the polynomial g(x) is given by:
[tex]g(x)=\dfrac{1}{4}(x^4-3x^3-12x^2+20x+48)[/tex]
