Question

Which trinomials are prime?


Choose all answers that are correct.

Answer 1

Answer:

$x^2+11x+18$ is prime

Step-by-step explanation:

Prime Trinomials

A given trinomial is prime if it cannot be factored. The general expression for a trinomial is

$ax^2+bx+c$

Another useful expression is

$a(x-x_1)(x-x_2)$

where x1 and x2 are the zeroes of the trinomial.

The zeroes can be computed by using the known formula

$\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

The trinomial has two zeroes if the root can be found as a real number, that is if d is positive or zero, where

$d=b^2-4ac$

Let's test the four options

$x^2-8x+2$

$a=1,b=-8,c=2$

$d=(-8)^2-4\cdot 1\cdot 2=56$

Since d is positive, the trinomial can be factored and therefore is not prime

$x^2+4x+5$

$a=1,b=4,c=5$

$d=4^2-4\cdot 1\cdot 5=-4$

Since d is negative, the trinomial is prime

$x^2+11x+18$

$a=1,b=11,c=18$

$d=11^2-4\cdot 1\cdot 18=49$

Since d is positive, the trinomial can be factored and therefore is not prime

$x^2-3x-2$

$a=1,b=-3,c=-2$

$d=(-3)^2-4\cdot 1\cdot (-2)=17$

Since d is positive, the trinomial can be factored and therefore is not prime