Find all zeroes of the polynomial `2x^4 – 10x^3 + 5x^2 + 15x – 12` when its two zeroes are `sqrt(3/2)` and `-sqrt(3/2)`

Answer

To find all the zeroes of the polynomial \( P(x) = 2x^4 - 10x^3 + 5x^2 + 15x - 12 \) given that two of its zeroes are \( \sqrt{\frac{3}{2}} \) and \( -\sqrt{\frac{3}{2}} \), we can follow these steps:

### Step 1: Identify the given zeroes
The given zeroes are:
- \( \alpha = \sqrt{\frac{3}{2}} \)
- \( \beta = -\sqrt{\frac{3}{2}} \)

### Step 2: Form factors from the given zeroes
Since \( \alpha \) and \( \beta \) are zeroes of the polynomial, we can form factors:
- \( (x - \alpha) = \left(x - \sqrt{\frac{3}{2}}\right) \)
- \( (x - \beta) = \left(x + \sqrt{\frac{3}{2}}\right) \)

### Step 3: Multiply the factors to find a quadratic factor
We can multiply these two factors:
\[ (x - \sqrt{\frac{3}{2}})(x + \sqrt{\frac{3}{2}}) = x^2 - \left(\sqrt{\frac{3}{2}}\right)^2 = x^2 - \frac{3}{2} \]

### Step 4: Divide the polynomial by the quadratic factor
Now, we need to divide the polynomial \( P(x) \) by \( x^2 - \frac{3}{2} \) to find the other factors.

Using polynomial long division:
1. Divide the leading term \( 2x^4 \) by \( x^2 \) to get \( 2x^2 \).
2. Multiply \( 2x^2 \) by \( x^2 - \frac{3}{2} \) to get \( 2x^4 - 3x^2 \).
3. Subtract this from \( P(x) \):
\[ (2x^4 - 10x^3 + 5x^2 + 15x - 12) - (2x^4 - 3x^2) = -10x^3 + 8x^2 + 15x - 12 \]
4. Repeat the process with \( -10x^3 \):
- Divide \( -10x^3 \) by \( x^2 \) to get \( -10x \).
- Multiply \( -10x \) by \( x^2 - \frac{3}{2} \) to get \( -10x^3 + 15x \).
- Subtract:
\[ (-10x^3 + 8x^2 + 15x - 12) - (-10x^3 + 15x) = 8x^2 - 12 \]
5. Finally, divide \( 8x^2 - 12 \) by \( x^2 - \frac{3}{2} \):
- Divide \( 8x^2 \) by \( x^2 \) to get \( 8 \).
- Multiply \( 8 \) by \( x^2 - \frac{3}{2} \) to get \( 8x^2 - 12 \).
- Subtract:
\[ (8x^2 - 12) - (8x^2 - 12) = 0 \]

Thus, we have:
\[ P(x) = (x^2 - \frac{3}{2})(2x^2 - 10x + 8) \]

### Step 5: Factor the quadratic \( 2x^2 - 10x + 8 \)
Now we can factor \( 2x^2 - 10x + 8 \) using the middle-term splitting method:
1. Rewrite as \( 2x^2 - 8x - 2x + 8 \).
2. Factor by grouping:
\[ 2x(x - 4) - 2(x - 4) = (2x - 2)(x - 4) \]

### Step 6: Set each factor to zero
Now we have:
\[ P(x) = (x^2 - \frac{3}{2})(2x - 2)(x - 4) \]
Setting each factor to zero gives us:
1. \( x^2 - \frac{3}{2} = 0 \) leads to \( x = \sqrt{\frac{3}{2}}, -\sqrt{\frac{3}{2}} \)
2. \( 2x - 2 = 0 \) leads to \( x = 1 \)
3. \( x - 4 = 0 \) leads to \( x = 4 \)

### Final Zeroes
Thus, the zeroes of the polynomial \( P(x) \) are:
- \( \sqrt{\frac{3}{2}} \)
- \( -\sqrt{\frac{3}{2}} \)
- \( 1 \)
- \( 4 \)