In a polynomial p(x) = x2 + x + 41 put different values of x and find

1.	In a polynomial p(x) = x2 + x + 41 put different values of x and find p(x). Can you conclude after putting different values of x that p(x) is prime for all. Is ‘x’ an element of N ? Put x = 41 in p(x). Now what do you find ?
Answer» p(x) = x² + x + 41 p(0) = 0² + 0 + 41 = 41 – is a prime p(1) = 1² + 1 + 41 = 43 – is a prime p(2) = 2² + 2 + 41 = 47 – is a prime p(3) = 3² + 3 + 41 = 53 – is a prime p(41) = 41² + 41 + 41 = 41(41 + 1 + 1) = 41 x 43 is not a prime. ∴ p(x) = x² + x + 41 is not a prime for all x. ∴ The conjecture “p(x) = x² + x + 41 is a prime” is false.

In a polynomial p(x) = x2 + x + 41 put different values of x and find p(x). Can you conclude after putting different values of x that p(x) is prime for all. Is ‘x’ an element of N ? Put x = 41 in p(x). Now what do you find ?

Answer»

p(x) = x² + x + 41

p(0) = 0² + 0 + 41 = 41 – is a prime

p(1) = 1² + 1 + 41 = 43 – is a prime

p(2) = 2² + 2 + 41 = 47 – is a prime

p(3) = 3² + 3 + 41 = 53 – is a prime

p(41) = 41² + 41 + 41

= 41(41 + 1 + 1)

= 41 x 43 is not a prime.

∴ p(x) = x² + x + 41 is not a prime for all x.

∴ The conjecture “p(x) = x² + x + 41 is a prime” is false.