Question

A prime polynomial is an irreducible polynomial with integer coefficients that cannot be factored into polynomials of lower degree over the real number system.

Which trinomial is prime?

x2 – 7x – 18
x2 – 9x – 18
x2 + 3x – 18
x2 + 17x – 18

Answer 1

Step-by-step explanation:

the point is for such a consideration :

the constant term (like -18 in our example here) is a×b.

and the factor of the x-term (like -7 of -7x) is a+b.

if there are 2 numbers a, b that satisfy these 2 conditions for a trinomial, than it is not prime as it can be factored.

x² - 7x - 18 is not prime.

what factors can create 18 ?

1×18

2×9

3×6

4×4.5

5×3.6

...

-7 = -9 + 2

so, it can be factored into

(x - 9)(x + 2)

x² - 9x - 18 is not prime

a×b = -18

a+b = -9

a = -9 - b

(-9 - b)b = -18

-9b - b² = -18

b² + 9b = 18

b² + 9b - 18 = 0

x² + 9b - 18 = 0

general solution to a quadratic equation :

x = (-b ± sqrt(b² - 4ac))/(2a) = (-9 ± sqrt(9² - 4×1×-18))/(2×1) =

= (-9 ± sqrt(81 + 72))/2 = (-9 ± sqrt(153))/2

x1 = b1 = (-9 + sqrt(153))/2 = 1.684658438... = a2

x2 = b2 = (-9 - sqrt(153))/2 = -10.68465844... = a1

so, we can factor this into

(x + 1.684658438...)(x - 10.68465844...)

or

(x + (-9 + sqrt(153))(x + (-9 - sqrt(153))

these are not rational but still real numbers.

x² + 3x - 18 is not prime

based on the previous 2 examples we can factor this

(x + 6)(x - 3)

x² + 17x - 18 is not prime

based on the same principles this can be factored

(x + 18)(x - 1)

Answer 2

The prime trinomial is x² - 7x - 18.

Step-by-step explanation:

To determine the prime trinomial, we need to check if any of them can be factored into polynomials of lower degree using integer coefficients. We can try factoring by grouping or using the quadratic formula to find the roots and see if they are integers.

x² - 7x - 18: This factors neatly as (x - 9)(x + 2), which means it is not prime.

x² - 9x - 18: This factors as (x - 9)(x + 2) as well, so it is not prime.

x² + 3x - 18: This factors as (x - 3)(x + 6), again not prime.

x² + 17x - 18: This trinomial has no integer roots when applying the quadratic formula, making it difficult to factor. Additionally, its coefficient of the x² term is 1, which is typically a characteristic of prime trinomials.

Therefore, based on the analysis of factors and coefficient of the x² term, x² + 17x - 18 is the most likely candidate for a prime trinomial. Further mathematical proofs or factoring techniques could be used to confirm its primeness definitively.