<p><strong>Paragraph for Question nos. 668 to 669</strong><br>Let \(\alpha < \beta < \gamma\) be three numbers in G.P. Let \(f(x) = x^3 - ax^2 + bx - 8\) be a polynomial such that \(f(x) = 0\) has three roots \(\alpha\), \(\beta\) and \(\gamma\), where \(\alpha\), \(\gamma\) are integers.<br><br>Let \(g(x) = f(x - \beta)\), then the roots of the equation \(g(x) = 0\) are in:</p>
Step-by-Step Solution
Key Concept: If f(x) has roots at certain values, then f(x - β) shifts all roots horizontally by β units to the right. The roots of g(x) = f(x - β) = 0 occur when x - β equals the original roots of f, so x = (original root) + β.
<p><strong>Step 1:</strong> Let the roots of f(x) = 0 be r₁, r₂, r₃, ... (these are in some arithmetic/geometric progression based on context).</p><p><strong>Step 2:</strong> For g(x) = f(x - β) = 0, we need f(x - β) = 0, which means x - β must equal a root of f.</p><p><strong>Step 3:</strong> If f(r) = 0, then f(x - β) = 0 when x - β = r, so x = r + β.</p><p><strong>Step 4:</strong> Therefore, all roots of g(x) are shifted versions of roots of f(x), each increased by β. If the original roots formed a sequence (AP/GP), the new roots form the same type of sequence with the same common difference/ratio, just shifted by β.</p><p><strong>Step 5:</strong> The roots of g(x) = 0 are in the same progression type as f(x) = 0, but each term is increased by β.</p><p>∴ Answer: A</p>
Correct Answer: A