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

`{x}^(3) + 2 {x}^(2) - 34x + 7`

Explanation:

Since the zeros are 4,1 and -7

We say

`(x - 4)(x - 1)(x + 7)`

Are the equations

Expanding this we get

`{x}^(3) + 2 {x}^(2) - 34x + 7`

Answer 2

`\large\text{$f(x)=\boxed{1}\;x^3+\boxed{2}\;x^2+\boxed{-31}\;x+\boxed{28}$}`

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 = 4`
• 
  `x = 1`
• 
  `x = -7`

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

• 
  `(x - 4)`
• 
  `(x - 1)`
• 
  `(x - (-7))=(x+7)`

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

`f(x)=(x-4)(x-1)(x+7)`

`f(x)=(x^2-x-4x+4)(x+7)`

`f(x)=(x^2-5x+4)(x+7)`

`f(x)=x^3+7x^2-5x^2-35x+4x+28`

`f(x)=x^3+2x^2-31x+28`

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

`\large\text{$f(x)=\boxed{1}\;x^3+\boxed{2}\;x^2+\boxed{-31}\;x+\boxed{28}$}`