IF `alpha,beta` are the zeroes of the polynomial `5x^2-7x+2` then the sum of their reciprocal is:

Options

A. `7/2`
B. `7/5`
C. `2/5`
D. `14/25`

Answer

To find the sum of the reciprocals of the zeros (roots) of the polynomial \(5x^2 - 7x + 2\), we will follow these steps:

### Step 1: Identify the coefficients
The given polynomial is \(5x^2 - 7x + 2\). Here, we can identify the coefficients:
- \(a = 5\) (coefficient of \(x^2\))
- \(b = -7\) (coefficient of \(x\))
- \(c = 2\) (constant term)

**Hint:** Coefficients are the numerical factors in front of the variable terms in the polynomial.

### Step 2: Use the formulas for the sum and product of roots
For a quadratic polynomial \(ax^2 + bx + c\), the sum of the roots \((\alpha + \beta)\) and the product of the roots \((\alpha \beta)\) can be calculated using the formulas:
- Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
- Product of roots: \(\alpha \beta = \frac{c}{a}\)

**Hint:** Remember that the sum of roots is derived from the negative coefficient of \(x\) divided by the coefficient of \(x^2\).

### Step 3: Calculate the sum of the roots
Using the formula for the sum of the roots:
\[ \alpha + \beta = -\frac{-7}{5} = \frac{7}{5} \]

**Hint:** Pay attention to the signs when applying the formula.

### Step 4: Calculate the product of the roots
Using the formula for the product of the roots:
\[ \alpha \beta = \frac{2}{5} \]

**Hint:** The product of the roots is simply the constant term divided by the coefficient of \(x^2\).

### Step 5: Find the sum of the reciprocals
The sum of the reciprocals of the roots can be expressed as:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} \]
Substituting the values we found:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\frac{7}{5}}{\frac{2}{5}} = \frac{7}{5} \times \frac{5}{2} \]

### Step 6: Simplify the expression
Now simplifying:
\[ \frac{7 \cdot 5}{5 \cdot 2} = \frac{7}{2} \]

### Final Answer
Thus, the sum of the reciprocals of the zeros of the polynomial \(5x^2 - 7x + 2\) is:
\[ \frac{7}{2} \]

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