Question

which polynomial is prime? $x^2 - 36$ $x^2 - 7x + 12$ $x^2 - x - 20$ $x^2 + 6$

Explanation:

Step1: Factor $x^2-36$ (difference of squares)

$x^2-36=(x-6)(x+6)$

Step2: Factor $x^2-7x+12$

Find two numbers that multiply to 12 and add to -7: -3 and -4.
$x^2-7x+12=(x-3)(x-4)$

Step3: Factor $x^2-x-20$

Find two numbers that multiply to -20 and add to -1: -5 and 4.
$x^2-x-20=(x-5)(x+4)$

Step4: Check $x^2+6$ for factoring

No real integers multiply to 6 and add to 0 (coefficient of $x$ is 0). It cannot be factored into linear terms with real integer coefficients, so it is prime.

Answer:

$\boldsymbol{x^2+6}$