Find a quadratic polynomial each with the given numbers as the sum and product of the zeros respectively. (i) `2/3, -1/3`, (ii) `0,-4sqrt(3)`, (iii) `-3/(2sqrt(5)), -1/2`, (iv) `21/8, 5/16`

Answer

To find a quadratic polynomial given the sum and product of its zeros, we can use the standard form of a quadratic polynomial:

\[ P(x) = x^2 - (sum \ of \ zeros) \cdot x + (product \ of \ zeros) \]

Let's solve the problem step by step for each part.

### (i) Given: Sum of zeros = \( \frac{2}{3} \), Product of zeros = \( -\frac{1}{3} \)

1. **Identify the sum and product of the zeros**:
- \( \alpha + \beta = \frac{2}{3} \)
- \( \alpha \beta = -\frac{1}{3} \)

2. **Substitute into the polynomial formula**:
\[ P(x) = x^2 - \left(\frac{2}{3}\right)x + \left(-\frac{1}{3}\right) \]

3. **Simplify the polynomial**:
\[ P(x) = x^2 - \frac{2}{3}x - \frac{1}{3} \]

4. **Multiply through by 3 to eliminate fractions**:
\[ P(x) = 3x^2 - 2x - 1 \]

### (ii) Given: Sum of zeros = \( 0 \), Product of zeros = \( -4\sqrt{3} \)

1. **Identify the sum and product of the zeros**:
- \( \alpha + \beta = 0 \)
- \( \alpha \beta = -4\sqrt{3} \)

2. **Substitute into the polynomial formula**:
\[ P(x) = x^2 - (0)x + (-4\sqrt{3}) \]

3. **Simplify the polynomial**:
\[ P(x) = x^2 - 4\sqrt{3} \]

### (iii) Given: Sum of zeros = \( -\frac{3}{2\sqrt{5}} \), Product of zeros = \( -\frac{1}{2} \)

1. **Identify the sum and product of the zeros**:
- \( \alpha + \beta = -\frac{3}{2\sqrt{5}} \)
- \( \alpha \beta = -\frac{1}{2} \)

2. **Substitute into the polynomial formula**:
\[ P(x) = x^2 - \left(-\frac{3}{2\sqrt{5}}\right)x + \left(-\frac{1}{2}\right) \]

3. **Simplify the polynomial**:
\[ P(x) = x^2 + \frac{3}{2\sqrt{5}}x - \frac{1}{2} \]

4. **Multiply through by \( 2\sqrt{5} \) to eliminate fractions**:
\[ P(x) = 2\sqrt{5}x^2 + 3x - \sqrt{5} \]

### (iv) Given: Sum of zeros = \( \frac{21}{8} \), Product of zeros = \( \frac{5}{16} \)

1. **Identify the sum and product of the zeros**:
- \( \alpha + \beta = \frac{21}{8} \)
- \( \alpha \beta = \frac{5}{16} \)

2. **Substitute into the polynomial formula**:
\[ P(x) = x^2 - \left(\frac{21}{8}\right)x + \left(\frac{5}{16}\right) \]

3. **Simplify the polynomial**:
\[ P(x) = x^2 - \frac{21}{8}x + \frac{5}{16} \]

4. **Multiply through by 16 to eliminate fractions**:
\[ P(x) = 16x^2 - 42x + 5 \]

### Final Answers:
1. \( P(x) = 3x^2 - 2x - 1 \)
2. \( P(x) = x^2 - 4\sqrt{3} \)
3. \( P(x) = 2\sqrt{5}x^2 + 3x - \sqrt{5} \)
4. \( P(x) = 16x^2 - 42x + 5 \)