Use the Rational Zeros Theorem to find all the real zeros

Use the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.

f(x)=3x^4 - 7x^3 + 13x^2 - 21x + 12

I don't know how to do this and I would like to know the steps to figure it out, I do know the answer is
1, 3; f(x)= (x-1)(3x-4)(x^2+3)

thank you!

The rational roots theorem says that IF there is a rational root of a polynomial f with interger coefficients, and a/b is such a root, in lowest terms (a and b have no common factors), then a divides the constant term, and b divides the coefficient of the largest power of x.

So any rational root of f = a/b would require that a divide 12 (meaning a could only be 1, 2, 3, 4, 6 or 12), and b would have to divide 3, meaning it could only be 1 or 3. Oh yeah, it can be positive or negative too. So technically I should have included -1 -2 -3 etc. in the possibilities for a.

From there, you have to start testing the possibilities by picking a and b from the list of possibilities, and plugging them in. It's usually easiest to start with the integers, especially since 1 turns out to be a root.

That can be a lot of work, graphing the function can help focus the search. Eventually you would find that 4/3 is a root, meaning (x - 4/3) divides f. You'd multiply by 3 to get integer coefficients, which is where (3x-4) comes from. After you divide f by (x-1) and (4x-3), you're left with x^2+3 (I assume your answer is correct--I didn't check it), which has no real roots -- obviously the other 2 roots are +/- √3.