Question

If $\alpha, \beta$ are the zeroes of the polynomial $p(x) = x^2 - 3x - 1$, then find the value of $\frac{1}{\alpha} + \frac{1}{\beta}$.

Hint:

Never waste time trying to find the actual roots of the quadratic equation using the quadratic formula unless specifically asked.
Most symmetric expressions in $\alpha$ and $\beta$ can be easily rewritten in terms of $(\alpha + \beta)$ and $(\alpha\beta)$.
For instance:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]

Answer 1

Step 1: Understanding the Question:
The topic of this question is Polynomials, specifically the relationship between the zeroes and coefficients of a quadratic polynomial.
We are given a quadratic polynomial, $p(x) = x^2 - 3x - 1$, with zeroes $\alpha$ and $\beta$.
We need to determine the value of the algebraic expression $\frac{1}{\alpha} + \frac{1}{\beta}$ without necessarily finding the individual values of $\alpha$ and $\beta$.

Step 2: Key Formula or Approach:
For a standard quadratic polynomial $ax^2 + bx + c$ with zeroes $\alpha$ and $\beta$, the relationships are:

• Sum of zeroes: $\alpha + \beta = -\frac{b}{a}$

• Product of zeroes: $\alpha\beta = \frac{c}{a}$

We can algebraically simplify the expression we want to evaluate:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha\beta} = \frac{\alpha + \beta}{\alpha\beta} \]
This allows us to substitute the sum and product of the zeroes directly into the simplified expression.

Step 3: Detailed Explanation:

• Identify the coefficients of the given quadratic polynomial $p(x) = x^2 - 3x - 1$ by comparing it with $ax^2 + bx + c$:
$a = 1$
$b = -3$
$c = -1$

• Compute the sum of the zeroes ($\alpha + \beta$):
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-3}{1} = 3 \]

• Compute the product of the zeroes ($\alpha\beta$):
\[ \alpha\beta = \frac{c}{a} = \frac{-1}{1} = -1 \]

• Express the required term $\frac{1}{\alpha} + \frac{1}{\beta}$ by taking a common denominator:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} \]

• Substitute the values of $(\alpha + \beta) = 3$ and $(\alpha\beta) = -1$ into this expression:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{3}{-1} = -3 \]


Step 4: Final Answer:
The value of $\frac{1}{\alpha} + \frac{1}{\beta}$ is $-3$.