Answer:
x = 1/3, x = -3, x = 0 and x = 3
Explanation:
Note that x can be factored out of f(x)=3x^4−x^3−27x^2+9x immediately:
f(x) = x(3x^3 - x^2 - 27x + 9.
Let's guess at the zeros and use synthetic div. to determine whether our guess actually is a zero:
Is 3 (a factor of 9) a zero? Use 3 as a divisor in synth. div.:
3 / 3 -1 -27 9
9 24 -9
-------------------------
3 8 -3 0
Because the remainder is zero, 3 is a zero of the given polynomial. So is 0 (which we know from having factored x out of the given f(x)=3x^4−x^3−27x^2+9x). The quotient is 3x^2 + 8x - 3, using the coefficients derived thru synth. div., above.
Let's use the quadratic formula to find the zeros of 3x^2 + 8x - 3:
a = 3, b = 8, c = -3
Then the discriminant is b^2 - 4(a)(c) = 8^2 - 4(3)(-3) = 64 + 36 = 100, and the square root of that is 10.
Thus, the zeros of 3x^2 + 8x - 3 are:
-8 plus or minus 10
x = -------------------------------
2(3)
or x = 1/3 and x = -3