can you factor this trinomial?

Question

x^2-x-462

6x^2+17x+10

or is it prime?

Answer 1

It is prime. If it was factorable, you could write it as (x -a)(x+b). then to get -x, you'd have to have a + b = -1, or a = b -1. So a and b are only one apart. Then one of them must be odd and the other one even. An even times an odd is odd. So ab can't be 462 ( an even number)

Answer 2

Those are binomials, not trinomials. Also, there is no such thing as a "prime" polynomial... I assume you mean a polynomial with no real roots, in which case you say "the roots are imaginary".

Use the quadratic formula or complete the square.

x² - x - 462 = 0
x² - x = 462
x² - x + (1/2)² = (1/2)² + 462
(x + 1/2)² = 1849/4
x + 1/2 = ± sqrt(1849/4)
x = -1/2 ± sqrt(1849)/2
x = -(1 ± 43)/2
x = -22, 21

Factored form:
(x + 22)(x - 21)


Similar deal for the second:
x² + (17/6)x + (10/6) = 0
x² + (17/6)x = -5/3
x² + (17/6)x + (17/12)² = (17/12)² - 5/3
(x + 17/12)² = 289/144 - 240/144
x + 17/12 = ±sqrt(49/144)
x = -17/12 ± 7/12
x = -(17 ± 7)/12
x = -12/6, -5/6

Factored form:
6(x + 12/6)(x + 5/6)

Answer 3

you could use the quadratic formula:

(-b +(-) sgrt(b^2 - 4ac)) / (2a)