To find a quadratic polynomial whose zeroes are -7 and [tex]\(-\frac{1}{7}\)[/tex], we can follow these steps:
### Step 1: Identify the Zeroes
Let the zeroes of the polynomial be [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex]. Here, they are:
[tex]\[\alpha = -7 \][/tex]
[tex]\[\beta = -\frac{1}{7} \][/tex]
### Step 2: Calculate the Sum of the Zeroes
The sum of the zeroes [tex]\((\alpha + \beta)\)[/tex] will be calculated as follows:
[tex]\[ \alpha + \beta = -7 + \left(-\frac{1}{7}\right) \][/tex]
From the given answer, the sum of the zeroes is:
[tex]\[ \alpha + \beta = -7.142857142857143 \][/tex]
### Step 3: Calculate the Product of the Zeroes
The product of the zeroes [tex]\((\alpha \times \beta)\)[/tex] will be calculated as:
[tex]\[ \alpha \cdot \beta = -7 \cdot \left(-\frac{1}{7}\right) \][/tex]
From the given answer, the product of the zeroes is:
[tex]\[ \alpha \cdot \beta = 1.0 \][/tex]
### Step 4: Form the Quadratic Polynomial
The standard form of a quadratic polynomial with zeroes [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] is:
[tex]\[ x^2 - (\alpha + \beta)x + \alpha \beta \][/tex]
Plugging in the values calculated for the sum and product of zeroes:
[tex]\[ x^2 - (-7.142857142857143)x + 1.0 \][/tex]
### Step 5: Write the Polynomial in Standard Form
Thus, the quadratic polynomial satisfying the given zeroes is:
[tex]\[ x^2 + 7.142857142857143x + 1.0 \][/tex]
This polynomial has the zeroes [tex]\(-7\)[/tex] and [tex]\(-\frac{1}{7}\)[/tex] as required.
