Polynomials of Real Coefficients and Multiplicity

Polynomials of Real Coefficients and Multiplicity

The equation \( 4x^3-27x+k=0 \) has a double root. Find the possible values of \( k \).

When we differentiate a polynomial function, we get its derivative, which provides information about the slope and behaviour of the original function. In particular, for a polynomial function, the multiplicity of a root can be determined by looking at the behaviour of the derivative at that root.

  1. If \( P(\alpha) \ne 0 \) and \( P^{\prime}(\alpha) \ne 0 \): In this case, the root at \( x = \alpha \) is a single root or has a multiplicity of \( 1 \). The graph of the function crosses the \(x\)-axis at \( x = \alpha \), and the derivative is nonzero at that point. The function behaves like a straight line passing through the \(x\)-axis at that point.
  2. If \( P(\alpha)= 0 \) and \( P^{\prime}(\alpha) \ne 0 \): Here, the root at \( x = \alpha \) has a multiplicity greater than \( 1 \). The graph of the function touches the \(x\)-axis at \( x = \alpha \) but doesn’t cross it. The derivative is still nonzero at that point, indicating that the function has a flat tangent line at that root.
  3. If \( P(\alpha) = P^{\prime}(\alpha) = 0 \): In this case, the root at \( x = \alpha \) has a multiplicity greater than \( 1 \), and the graph of the function also touches the \(x\)-axis at \( x = \alpha \). However, unlike the previous case, the derivative is also zero at that point, indicating that the function has a “bouncing” behaviour near the root.

If \( P(x) \) is a polynomial with real coefficients, has a root \( \alpha \) of multiplicity \( r \), then \( P^{\prime} (x) \) has the root \( \alpha \) with multiplicity \( r-1 \).

\( \displaystyle \begin{align} P(x) &= 4x^3-27x+k \\ P^{\prime} (x) &= 12x^2-27 \end{align} \)

If \( P(x) \) has a double root \( \alpha \), then \( \alpha \) is a single root of \( P^{\prime}(x) \).

\( \displaystyle \begin{align} P^{\prime}(\alpha) &= 12 \alpha^2-27 \\ 12 \alpha^2-27 &= 0 \\ \alpha^2 &= \frac{9}{4} \\ \alpha &= \pm \frac{3}{2} \end{align} \)

Thus \( \displaystyle \alpha = \frac{3}{2} \) or \( \displaystyle \alpha = -\frac{3}{2} \) is a double root of \( P(x) = 0 \).

\( \displaystyle \begin{align} 4 \left( \frac{3}{2} \right)^3-27 \left( \frac{3}{2} \right) + k &= 0 \\ -27 + k &= 0 \\ \therefore k &= 27 \\ 4 \left( -\frac{3}{2} \right)^3-27 \left( -\frac{3}{2} \right) + k &= 0 \\ 27 + k &= 0 \\ \therefore k &= -27 \end{align} \)

Thus, the possible values of \( k=\pm 27 \).

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