Can anybody help me solve this problem 3x^4-11x^3-3x^2-6x+8=0?

Question

Can somebody help me find all zeros of the polynomial functions or solve the given polynomial equation. Use the rational zero theorem, describe rule of signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root.

Answer 1

Rational zeros theorem tells us any rational zeros must be (±1, ±2, ±4, ±8) / (1, 3)
Rule of signs tells us that there are 0 or 2 positive roots and 0 or 2 negative roots.
It's not too hard to see that two roots are 4 and 2/3. So we get
3x^4 - 11x^3 - 3x^2 - 6x + 8
= (3x - 2) (x^3 - 3x^2 - 3x - 4)
= (3x - 2) (x - 4) (x^2 + x + 1)
and x^2 + x + 1 is irreducible, so the only zeros are 4 and 2/3.

Answer 2

rational zeros say possibles are
+-1,2,4,8
------------
+- 1,3

so try +- 1,1/3,2,2/3,4,4/3,8,8/3
Look at the graph for the x ints