Find all rational zeros of the function. Then (if necessary) use the depressed equation to find all roots of the equation f(x)=0

f(x)= 5x^4-11x^3-33x^2+77x-14

trying x = ±1, ±2, ± 7, ±1/5, ±2/5, ±7/5

I found f(2) = 0
and f(x) = (x-2)(5x^3 - x^2 - 35x + 7) = (x-2)g(x)
x = ±7 didn't work in the first, so it certainly will not work in this one
so try x = 1/5 in the cubic factor
g(1/5) = 5(1/125) - 1/25 - 35(1/5) + 7 = 0

so we have
f(x) = (x-2)(5x-1)(x^2 - 7) after a division

roots are
x = 2, 1/5, ± √7

To find the rational zeros of a polynomial function, you can use the Rational Root Theorem. According to the theorem, the possible rational roots are all the ratios ± p/q, where p is a factor of the constant term (-14 in this case) and q is a factor of the leading coefficient (5 in this case).

1. Factors of -14: ±1, ±2, ±7, ±14
2. Factors of 5: ±1, ±5

So, the possible rational zeros are: ±1/1, ±2/1, ±7/1, ±14/1, ±1/5, ±2/5, ±7/5, ±14/5

Now, let's use these possible rational zeros in synthetic division to check if they are actual zeros:

1. Trying x = 1:
1 | 5 -11 -33 77 -14
| 5 -6 -39 38
-----------------------------
5 -6 -39 38 24

2. Trying x = -1:
-1 | 5 -11 -33 77 -14
| -5 16 17 -60
-----------------------------
5 -16 -17 94 -74

3. Trying x = 2:
2 | 5 -11 -33 77 -14
| 10 -2 -62 30
-----------------------------
5 -1 -35 15 16

4. Trying x = -2:
-2 | 5 -11 -33 77 -14
| -10 42 -18 -58
-----------------------------
5 -21 9 59 -72

5. Trying x = 7:
7 | 5 -11 -33 77 -14
| 35 168 1052 6091
-----------------------------
5 24 111 612 6077

Now, we check the remainders from the synthetic divisions. If the remainder is 0 (or close to 0 due to rounding errors), it means that the corresponding x-value is a zero.

From the synthetic divisions, we see that x = 2 and x = -1 are actual rational zeros of the function f(x).

Next, we form the depressed equation by dividing f(x) by (x - 2)(x + 1). It gives:

f(x) = (x - 2)(x + 1)(5x^2 + 3x - 7)

So, the remaining part is the quadratic equation 5x^2 + 3x - 7. To find its roots, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Applying the formula, we get:

x = (-3 ± √(3^2 - 4 * 5 * -7)) / (2 * 5)
x = (-3 ± √(9 + 140)) / 10
x = (-3 ± √149) / 10

Therefore, the full list of roots for the equation f(x) = 0 is:
x = -1, 2, (-3 + √149) / 10, (-3 - √149) / 10