Question

Which of the following is a factor of f(x) = 5x3 + 24x2 − 75x + 14?
(x + 1 over 2)
(x − 1 over 2)
(x + 2)
(x − 2)

Answer 1

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The best and most correct answer among the choices provided by the question is (x − 2).       
    
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Answer 2

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

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