Question

If \(\alpha\) and \(\beta\) be the zeros of the polynomial \(cx^2 + ax + b\) then the value of \(\alpha . \beta\) is

• A. \(\frac{a}{c}\)
• B. \(-\frac{a}{c}\)
• C. \(\frac{b}{c}\)
• D. \(-\frac{b}{c}\)

Hint:

Be careful with the standard form. The question uses \(cx^2+ax+b\) instead of the usual \(ax^2+bx+c\). Always identify the coefficients based on the powers of x, not on the letters used.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For a general quadratic polynomial of the form \(Px^2 + Qx + R\), the product of its zeros is given by the ratio of the constant term to the coefficient of the \(x^2\) term.

Step 2: Key Formula or Approach:
For a polynomial \(Px^2 + Qx + R\), the product of zeros \(\alpha \beta = \frac{R}{P}\).

Step 3: Detailed Explanation:
The given polynomial is \(cx^2 + ax + b\).
Here, the coefficient of the \(x^2\) term is \(P = c\).
The coefficient of the \(x\) term is \(Q = a\).
The constant term is \(R = b\).
Applying the formula for the product of zeros:
\[ \alpha \beta = \frac{\text{constant term}}{\text{coefficient of } x^2} = \frac{b}{c} \]

Step 4: Final Answer:
The value of \(\alpha . \beta\) is \(\frac{b}{c}\).