Question

Question#21 : Write a polynomial function of least degree with integral coefficientd that had the given zeros.

Answer 1

From the question,

The zeros of the polynomial function are

`(2)/(3),-4,-(4)/(3)`

This implies that

`x=(2)/(3),x=-4,x=-(4)/(3)`

Hence we have

`x-(2)/(3)=0,x+4=0,x+(4)/(3)=0`

Hemce, the function f(x) will be

`f(x)=(x-(2)/(3))(x+4)(x+(4)/(3))`

Next, we are to expand the polynomial

This gives

`\begin{gathered} f(x)=(x^2+4x-(2)/(3)x-(8)/(3))(x+(4)/(3)) \\ f(x)=(x^2+(10)/(3)x-(8)/(3))(x+(4)/(3)) \end{gathered}`

Next we have

`\begin{gathered} f(x)=x^3+(4)/(3)x^2+(10)/(3)x^2+(40)/(9)x-(8)/(3)x-(32)/(9) \\ f(x)=x^3+(14)/(3)x^2+(16)/(9)x-(32)/(9) \\ f(x)=9x^3+42x^2+16x-32 \end{gathered}`

Therefore, the polynomial function is

`f(x)=9x^3+42x^2+16x-32`