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

Solution:

If the alternatives are factors of the given equation then they can be equated to zero and the value of x obtained when substituted will make the function zero.

Let us consider the first alternative i.e. x + 1/2 and if it is a factor then,

x + 1/2 = 0

x = -1/2

Substituting this value of x in the given equation f(x) = 5x³ + 24x² - 75x + 14 we obtain:

5(-1/2)³ + 24(-1/2)² + -75(-1/2) + 14

= -5/8 + 24(1/4) + 75/2

= - 5/8 + 6 + 37.5 = 36.875 which is not equal to zero.

Therefore x + 1/2 is not a factor

b) Now consider the second factor x - 1/2 . If it is factor then

x - 1/2 = 0

x = 1/2

Substituting this value of x in the given equation f(x) = 5x³ + 24x² - 75x + 14 we obtain:

5(1/2)³ + 24(1/2)² - 75(1/2) + 14

= 5/8 + 6 - 37.5 = 32.125 which is again not equal to zero.

Therefore x - 1/2 is not a factor

c) let us consider alternative 3 as the factor i.e. x + 2. If x + 2 is a factor then it can be equated to zero:

x + 2 = 0

x = -2

Substituting the above value in the given equation we get

5(-2)³ + 24(-2)² - 75(-2) + 14

=- 40 + 96 + 150 + 14 = 220 which is not zero, hence x + 2 is not a factor

d) Let us look at the last alternative x - 2. If x - 2 is a factor then it could be equated to zero. Therefore

x - 2 = 0

x = 2

Substituting the above value in the given equation we get

5(2)³ + 24(2)² - 75(2) + 14

= 40 + 96 - 150 + 14 = 0

Hence (x - 2) is the factor of the given equation.

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

Summary:

The factor of f(x) = 5x³ + 24x² - 75x + 14 is (x - 2). Hence last option is the correct answer.