Obtain all zeros of `(3x^4 -15x^3 + 13x^2 +25x -30)`, if two of its zeros are`sqrt(5/3) and - sqrt(5/3)`.

Explanation

To find all the zeros of the polynomial \( p(x) = 3x^4 - 15x^3 + 13x^2 + 25x - 30 \), given that two of its zeros are \( \sqrt{\frac{5}{3}} \) and \( -\sqrt{\frac{5}{3}} \), we can follow these steps:

### Step 1: Form a quadratic factor from the known zeros
Since we know two of the zeros, we can express them as factors:
\[ (x - \sqrt{\frac{5}{3}})(x + \sqrt{\frac{5}{3}}) = x^2 - \left(\sqrt{\frac{5}{3}}\right)^2 = x^2 - \frac{5}{3} \]
To eliminate the fraction, we can multiply the entire factor by 3:
\[ 3(x^2 - \frac{5}{3}) = 3x^2 - 5 \]
So, we have one factor of the polynomial:
\[ d(x) = 3x^2 - 5 \]

### Step 2: Use polynomial long division
Now we will divide the original polynomial \( p(x) \) by \( d(x) \):
\[ p(x) = 3x^4 - 15x^3 + 13x^2 + 25x - 30 \]
We will divide by \( 3x^2 - 5 \).

1. Divide the leading term: \( \frac{3x^4}{3x^2} = x^2 \).
2. Multiply \( x^2 \) by \( 3x^2 - 5 \):
\[ x^2(3x^2 - 5) = 3x^4 - 5x^2 \]
3. Subtract this from \( p(x) \):
\[ (3x^4 - 15x^3 + 13x^2 + 25x - 30) - (3x^4 - 5x^2) = -15x^3 + 18x^2 + 25x - 30 \]

4. Repeat the process: Divide the leading term \( \frac{-15x^3}{3x^2} = -5x \).
5. Multiply \( -5x \) by \( 3x^2 - 5 \):
\[ -5x(3x^2 - 5) = -15x^3 + 25x \]
6. Subtract:
\[ (-15x^3 + 18x^2 + 25x - 30) - (-15x^3 + 25x) = 18x^2 - 30 \]

7. Divide again: \( \frac{18x^2}{3x^2} = 6 \).
8. Multiply \( 6 \) by \( 3x^2 - 5 \):
\[ 6(3x^2 - 5) = 18x^2 - 30 \]
9. Subtract:
\[ (18x^2 - 30) - (18x^2 - 30) = 0 \]

So, we have \( p(x) = (3x^2 - 5)(x^2 - 5x + 6) \).

### Step 3: Factor the quadratic \( x^2 - 5x + 6 \)
Now we need to factor \( x^2 - 5x + 6 \):
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 4: Find all zeros
Now we can write the complete factorization of \( p(x) \):
\[ p(x) = (3x^2 - 5)(x - 2)(x - 3) \]
Setting each factor to zero gives us the zeros:
1. From \( 3x^2 - 5 = 0 \):
\[ 3x^2 = 5 \implies x^2 = \frac{5}{3} \implies x = \pm \sqrt{\frac{5}{3}} \]
2. From \( x - 2 = 0 \):
\[ x = 2 \]
3. From \( x - 3 = 0 \):
\[ x = 3 \]

### Final Answer
The zeros of the polynomial \( p(x) \) are:
\[ \sqrt{\frac{5}{3}}, -\sqrt{\frac{5}{3}}, 2, 3 \]