For each of the following find a quadratic polynomial whose sum and product respectively of the zeroes are as given .Also find the zeroes of these polynomials by factorisation.
`(21)/(8),(5)/(16)`

Answer

To find a quadratic polynomial whose sum and product of the zeroes are given, we can use the following steps:

### Step 1: Identify the sum and product of the zeroes
Given:
- Sum of the zeroes (α + β) = \( \frac{21}{8} \)
- Product of the zeroes (αβ) = \( \frac{5}{16} \)

### Step 2: Write the standard form of the quadratic polynomial
The standard form of a quadratic polynomial based on the sum and product of its zeroes is:
\[ P(x) = x^2 - (sum \ of \ zeroes) \cdot x + (product \ of \ zeroes) \]

Substituting the values:
\[ P(x) = x^2 - \left(\frac{21}{8}\right)x + \left(\frac{5}{16}\right) \]

### Step 3: Eliminate fractions by multiplying through by the least common multiple (LCM)
The LCM of the denominators (8 and 16) is 16. Multiply the entire polynomial by 16 to eliminate the fractions:
\[ 16P(x) = 16x^2 - 16 \cdot \left(\frac{21}{8}\right)x + 16 \cdot \left(\frac{5}{16}\right) \]

Calculating each term:
- \( 16 \cdot \left(\frac{21}{8}\right) = 2 \cdot 21 = 42 \)
- \( 16 \cdot \left(\frac{5}{16}\right) = 5 \)

Thus, we have:
\[ 16P(x) = 16x^2 - 42x + 5 \]

### Step 4: Write the polynomial in standard form
Now, we can express the polynomial as:
\[ P(x) = 16x^2 - 42x + 5 \]

### Step 5: Factor the polynomial
To find the zeroes, we need to factor the polynomial \( 16x^2 - 42x + 5 \).

1. We look for two numbers that multiply to \( 16 \cdot 5 = 80 \) and add to \( -42 \).
2. The numbers are \( -40 \) and \( -2 \).

Now, we can rewrite the polynomial:
\[ 16x^2 - 40x - 2x + 5 \]

3. Group the terms:
\[ (16x^2 - 40x) + (-2x + 5) \]
4. Factor by grouping:
\[ 8x(2x - 5) - 1(2x - 5) \]
5. Factor out the common term:
\[ (2x - 5)(8x - 1) \]

### Step 6: Find the zeroes
Set each factor to zero:
1. \( 2x - 5 = 0 \) → \( x = \frac{5}{2} \)
2. \( 8x - 1 = 0 \) → \( x = \frac{1}{8} \)

### Conclusion
The zeroes of the polynomial \( 16x^2 - 42x + 5 \) are:
- \( x = \frac{5}{2} \)
- \( x = \frac{1}{8} \)