Answer:
Polynomial equation is, [tex]p=(x+2)(3x-1)(x-3)=3x^3-4x^2-17x+6[/tex]
Step-by-step explanation:
Given:
The zeros of the polynomial, [tex]p[/tex], are [tex] x=-2,\frac{1}{3}, 3[/tex].
Therefore,
[tex]x= -2\\(x+2)=0[/tex]
So, [tex](x+2)[/tex] is a factor of the polynomial.
[tex]x=\frac{1}{3}\\3x=1\\3x-1=0[/tex]
So, [tex](3x-1)[/tex] is a factor of the polynomial.
[tex]x=3\\x-3=0[/tex]
So, [tex](x-3)[/tex] is also a factor of the polynomial.
Therefore, a polynomial can be expressed in terms of the product of its factors.
Thus, [tex]p=(x+2)(3x-1)(x-3)[/tex]
Expanding [tex](x+2)\textrm{ and }(3x-1)[/tex], we get
[tex](x+2)(3x-1)=3x^2-x+6x-2=3x^2+5x-2[/tex]
Now, expanding [tex](3x^2+5x-2)\textrm{ and }(x-3)[/tex], we get
[tex](3x^2+5x-2)(x-3)=3x^3-9x^2+5x^2-15x-2x+6=3x^3-4x^2-17x+2[/tex]
Therefore, the polynomial could be [tex]p=(x+2)(3x-1)(x-3)=3x^3-4x^2-17x+6[/tex]
