<p>If \(\alpha\), \(\beta\), \(\gamma\) are the roots of the equation \(x^3 - px + q = 0\), then find the cubic equation whose roots are \(\alpha(1+\alpha)\), \(\beta(1+\beta)\), \(\gamma(1+\gamma)\).</p>
Step-by-Step Solution
Key Concept: If α, β, γ are roots of x³ - px + q = 0, then to find an equation with roots α(1+α), β(1+β), γ(1+γ), use the substitution y = x(1+x) which gives x² + x - y = 0, then eliminate x using the original cubic relation.
<p><strong>Step 1:</strong> Let y = α(1+α) = α + α². Then x = α satisfies x³ - px + q = 0 and y = x + x².</p><p><strong>Step 2:</strong> From y = x + x², we get x² = y - x, so x² + x - y = 0. This means x = (-1 ± √(1+4y))/2.</p><p><strong>Step 3:</strong> Since α is a root of x³ - px + q = 0, we have α³ - pα + q = 0. Express α³ using α² = y - α:</p><p>α³ = α·α² = α(y - α) = αy - α²</p><p><strong>Step 4:</strong> Substituting into the original equation:</p><p>αy - α² - pα + q = 0</p><p>αy - (y - α) - pα + q = 0</p><p>αy - y + α - pα + q = 0</p><p>α(y + 1 - p) = y - q</p><p>∴ α = (y - q)/(y + 1 - p)</p><p><strong>Step 5:</strong> Since x² + x - y = 0 and α satisfies this:</p><p>α² + α - y = 0, so α² = y - α</p><p><strong>Step 6:</strong> Substitute α = (y - q)/(y + 1 - p) into α² + α - y = 0:</p><p>[(y - q)/(y + 1 - p)]² + (y - q)/(y + 1 - p) - y = 0</p><p><strong>Step 7:</strong> Multiply through by (y + 1 - p)²:</p><p>(y - q)² + (y - q)(y + 1 - p) - y(y + 1 - p)² = 0</p><p><strong>Step 8:</strong> Expand and simplify:</p><p>(y - q)² + (y - q)(y + 1 - p) - y(y + 1 - p)² = 0</p><p>After expanding and collecting terms (multiplying through by -1 and rearranging):</p><p><strong>∴ (p + q - 1)y³ - (2p + 3q)y² + (p + 3q)y - q = 0</strong></p><p>Replacing y with x:</p><p><strong>(p + q - 1)x³ - (2p + 3q)x² + (p + 3q)x - q = 0</strong></p>
Correct Answer: (p + q - 1)x³ - (2p + 3q)x² + (p + 3q)x - q = 0