rational roots theorem?

Question

please help and explain?

find the roots of
1) 10x^4 + x^3 + 7x^2 + x -3 = 0

a. -3/5, 1/2; +/- i
b. 3/5, -1/2; +/- i
c. -2/3, 3/4; +/- i
d. -1/3, 3/4; +/- i


2) x^4 - 7x^2 + 12 = 0

Answer 1

1) 10x^4 + x^3 + 7x^2 + x - 3 = 0

By the rational roots theorem, the potential rational roots are the plus or minus the factors of the constant term divided by the factors of the leading coefficient. These are only potential roots; it may be that only some of them are roots; it may be that none are.

Potential rational roots are:
±(1, 3) / (1, 2, 5, 10)

So c and d are out. No root can be a third or fourth.

That leaves only two possibilities. Try them both and we have:

10x^4 + x^3 + 7x^2 + x - 3 = 0
(5x + 3)(2x - 1)(x + i)(x - i) = 0

a. -3/5, 1/2; ± i
__________

2) x^4 - 7x^2 + 12 = 0
(x² - 3)(x² - 4) = 0
(x - √3)(x + √3)(x - 2)(x + 2) = 0

x = ±√3, ±2