Find the quadratic polynomial sum and product of whose zeroes are -1 and -20 respectively. Also, find the zeroes of the polynomial so obtained

Answer

To find the quadratic polynomial whose sum and product of zeroes are -1 and -20 respectively, we can follow these steps:

### Step 1: Identify the Sum and Product of Zeroes
Given:
- Sum of zeroes (α + β) = -1
- Product of zeroes (α * β) = -20

### Step 2: Use the Standard Form of a Quadratic Polynomial
The standard form of a quadratic polynomial can be expressed as:
\[ P(x) = x^2 - (α + β)x + (α * β) \]

### Step 3: Substitute the Values
Substituting the values of the sum and product of zeroes into the polynomial:
\[ P(x) = x^2 - (-1)x + (-20) \]
\[ P(x) = x^2 + x - 20 \]

### Step 4: Factor the Polynomial
Now, we will factor the polynomial \( P(x) = x^2 + x - 20 \).

To factor, we need two numbers that multiply to -20 (the product) and add to +1 (the sum). The numbers that meet these criteria are +5 and -4.

So, we can rewrite the polynomial as:
\[ P(x) = (x + 5)(x - 4) \]

### Step 5: Find the Zeroes of the Polynomial
To find the zeroes, we set the polynomial equal to zero:
\[ (x + 5)(x - 4) = 0 \]

Setting each factor to zero gives us:
1. \( x + 5 = 0 \) → \( x = -5 \)
2. \( x - 4 = 0 \) → \( x = 4 \)

### Conclusion
The quadratic polynomial is:
\[ P(x) = x^2 + x - 20 \]
The zeroes of the polynomial are:
- \( x = -5 \)
- \( x = 4 \)

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