Find the polynomial with rational coefficients and whose roots are
`0,1,-3//2,5//2`

Answer

To find the polynomial with rational coefficients whose roots are \(0, 1, -\frac{3}{2}, \frac{5}{2}\), we can follow these steps:

### Step 1: Identify the Roots
The roots of the polynomial are given as:
- \( \alpha = 0 \)
- \( \beta = 1 \)
- \( \gamma = -\frac{3}{2} \)
- \( \delta = \frac{5}{2} \)

### Step 2: Write the Polynomial in Factor Form
The polynomial can be expressed in factor form as:
\[ P(x) = (x - \alpha)(x - \beta)(x - \gamma)(x - \delta) \]
Substituting the values of the roots:
\[ P(x) = (x - 0)(x - 1)\left(x + \frac{3}{2}\right)\left(x - \frac{5}{2}\right) \]
This simplifies to:
\[ P(x) = x(x - 1)\left(x + \frac{3}{2}\right)\left(x - \frac{5}{2}\right) \]

### Step 3: Simplify the Factors
First, simplify \( (x + \frac{3}{2})(x - \frac{5}{2}) \):
\[ (x + \frac{3}{2})(x - \frac{5}{2}) = x^2 - \frac{5}{2}x + \frac{3}{2}x - \frac{15}{4} \]
Combining like terms:
\[ = x^2 - \frac{2}{2}x - \frac{15}{4} = x^2 - x - \frac{15}{4} \]

### Step 4: Multiply by the Remaining Factor
Now, we multiply this result by \( x(x - 1) \):
\[ P(x) = x(x - 1)(x^2 - x - \frac{15}{4}) \]
First, expand \( x(x - 1) \):
\[ x(x - 1) = x^2 - x \]
Now, multiply \( (x^2 - x) \) by \( (x^2 - x - \frac{15}{4}) \):
\[ P(x) = (x^2 - x)(x^2 - x - \frac{15}{4}) = x^4 - x^3 - \frac{15}{4}x^2 - x^3 + x^2 + \frac{15}{4}x \]
Combining like terms:
\[ = x^4 - 2x^3 + \left(1 - \frac{15}{4}\right)x^2 + \frac{15}{4}x \]
Calculating \( 1 - \frac{15}{4} = \frac{4}{4} - \frac{15}{4} = -\frac{11}{4} \):
\[ P(x) = x^4 - 2x^3 - \frac{11}{4}x^2 + \frac{15}{4}x \]

### Step 5: Clear the Fraction
To eliminate the fraction, multiply the entire polynomial by 4:
\[ 4P(x) = 4x^4 - 8x^3 - 11x^2 + 15x \]

### Final Polynomial
Thus, the polynomial with rational coefficients is:
\[ P(x) = 4x^4 - 8x^3 - 11x^2 + 15x \]