The zeroes of a polynomial are -`2/3` and `3`. Find the polynomial.

Options

A. `x^2 - 7/3x - 2 = 0`
B. `3x^2 - 7x -6 = 0`
C. `x^2 - 7/3x = 2`
D. All of these

Answer

To find the polynomial given the zeroes -2/3 and 3, we can follow these steps:

### Step 1: Identify the Zeroes
The zeroes of the polynomial are given as:
- \( \alpha = -\frac{2}{3} \)
- \( \beta = 3 \)

### Step 2: Calculate the Sum of the Zeroes
The sum of the zeroes \( S \) is calculated as follows:
\[ S = \alpha + \beta = -\frac{2}{3} + 3 \]
To add these, we convert 3 into a fraction:
\[ 3 = \frac{9}{3} \]
Now, adding the fractions:
\[ S = -\frac{2}{3} + \frac{9}{3} = \frac{7}{3} \]

### Step 3: Calculate the Product of the Zeroes
The product of the zeroes \( P \) is calculated as:
\[ P = \alpha \cdot \beta = -\frac{2}{3} \cdot 3 \]
Calculating this gives:
\[ P = -2 \]

### Step 4: Form the Polynomial
Using the sum and product of the zeroes, we can form the polynomial. The general form of a quadratic polynomial with zeroes \( \alpha \) and \( \beta \) is:
\[ x^2 - Sx + P \]
Substituting the values of \( S \) and \( P \):
\[ x^2 - \frac{7}{3}x - 2 \]

### Step 5: Eliminate the Fraction
To eliminate the fraction, we can multiply the entire polynomial by 3:
\[ 3 \left( x^2 - \frac{7}{3}x - 2 \right) = 3x^2 - 7x - 6 \]

### Final Polynomial
Thus, the polynomial we are looking for is:
\[ 3x^2 - 7x - 6 \]

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