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MissPhiladelphia

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The best and most correct answer among the choices provided by the question is (x − 2).       
    
Hope my answer would be a great help for you.

InesWalston

Answer:

(x-2) is a factor of the function.

Step-by-step explanation:

Given function is,

\(f(x) = 5x^3+24x^2-75x+14\)

As when we equate any factor of a polynomial to zero, the value of x we get is the zero of the polynomial.

So equating all the options to 0, we get

  1. \(x+\dfrac{1}{2}=0\ \ \ \ \Rightarrow x=-\dfrac{1}{2}\)
  2. \(x-\dfrac{1}{2}=0\ \ \ \ \Rightarrow x=\dfrac{1}{2}\)
  3. \(x+2=0\ \ \ \ \Rightarrow x=-2\)
  4. \(x-2=0\ \ \ \ \Rightarrow x=2\)

If x is a zero of a polynomial, then after putting the value of x in the polynomial, it becomes 0.

So,

  1. \(f(-\dfrac{1}{2}) = 5(-\dfrac{1}{2}) ^3+24(-\dfrac{1}{2}) ^2-75(-\dfrac{1}{2}) +14=\dfrac{455}{8} eq 0\)
  2. \(f(\dfrac{1}{2}) = 5(\dfrac{1}{2}) ^3+24(\dfrac{1}{2}) ^2-75(\dfrac{1}{2}) +14=-\dfrac{135}{8} eq 0\)
  3. \(f(-2) = 5(-2)^3+24(-2)^2-75(-2)+14=220 eq 0\)
  4. \(f(2) = 5(2)^3+24(2)^2-75(2)+14=0\)

So, (x-2) is a factor of the function.

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