find the number of positive integer x for which f(x)=x^3-8x^2+20x-13,is a prime number - Maths - Relations and Functions

Question

find the number of positive integer x for which
f(x)=x^3-8x^2+20x-13,is a prime number

Ashwini Kumar · Accepted Answer

We have,fx=x3-8x2+20x-13=x3-x2-7x2+7x+13x-13=x2x-1-7xx-1+13x-1=x-1x2-7x+13fx will be prime, only if one of x-1 and x2-7x+13 is 1 and another is prime
Or, one of x-1 and x2-7x+13 is -1 and another is negative of prime
But for positive integer x minimum value of x-1 will be 0
Now,When x-1=1 ⇒ x=2When x=2⇒x2-7x+13=-1⇒fx=-1So, this is not possible
Now, when x2-7x+13=1⇒x2-7x+12=0⇒x-4x-3=0⇒x=4, 3When, x=4x-1=3, x2-7x+13=1 and fx=3When, x=3x-1=2, x2-7x+13=1 and fx=2Therefore, when x=4 or 3 fx is prime
Hence, required number of positive integers=2