Question

HELLP!!!!!!!!

"Write the simplest polynomial function using the three zeros given, where all coefficients are integers and the leading coefficient is positive. Enter your answers below."

Answer 1

`\large\text{$f(x)=\boxed{1}\;x^3+\boxed{-12}\;x^2+\boxed{27}\;x+\boxed{40}$}`

Explanation:

If x = a is a zero (root) of a polynomial function f(x), then (x - a) is a factor of the polynomial function.

Given zeros:

• 
  `x = 5`
• 
  `x = -1`
• 
  `x = 8`

Therefore, the factors of polynomial f(x) are:

• 
  `(x - 5)`
• 
  `(x - (-1)) = (x + 1)`
• 
  `(x - 8)`

Multiply these factors together to find the polynomial f(x):

`f(x)=(x-5)(x+1)(x-8)`

`f(x)=(x^2+x-5x-5)(x-8)`

`f(x)=(x^2-4x-5)(x-8)`

`f(x)=x^3-8x^2-4x^2+32x-5x+40`

`f(x)=x^3-12x^2+27x+40`

Therefore, the simplest polynomial function with the given zeros is:

`\large\text{$f(x)=\boxed{1}\;x^3+\boxed{-12}\;x^2+\boxed{27}\;x+\boxed{40}$}`