Problem 75 - Calculus Single Variable

Question

Let \(p(x)\) be a polynomial. We say that \(p\) has a root of order \(k\) at \(a\) if there is a polynomial \(q\) such that \(p(x)=\) \((x-a)^{k} q(x)\) and \(q(a) \neq 0 .\) Prove that \(p\) has a root of order 2 at \(x=a\) if and only if \(p(a)=0, p^{\prime}(a)=0,\) and \(p^{\prime \prime}(a) \neq 0\). (In general, \(p\) has a root of order \(k\) at \(a\) if and only if \(p(a)=p^{\prime}(a)=p^{(k-1)}(a)=0\) and \(\left.p^{(k)}(a) \neq 0 .\right)\)

Step-by-Step Solution

Answer

A polynomial has a root of order 2 at \(x=a\) if and only if \(p(a)=0\), \(p'(a)=0\), and \(p''(a) \neq 0\).

Step 1: Understand the Problem Requirements

We want to show that a polynomial \(p(x)\) has a root of order 2 at \(x=a\) if and only if certain conditions involving its derivatives at \(a\) are met. These conditions are: \(p(a)=0\), \(p'(a)=0\), and \(p''(a) eq 0\).

Step 2: Assume p(x) has a Root of Order 2

Assume \(p(x)\) has a root of order 2 at \(x=a\). By definition, this means there exists a polynomial \(q(x)\) such that \(p(x) = (x-a)^2 q(x)\) and \(q(a) eq 0\).

Step 3: Evaluate p(a) and p'(a)

Since \(p(x) = (x-a)^2 q(x)\), when \(x=a\), \(p(a) = (a-a)^2 q(a) = 0\). Differentiate \(p(x)\) to get \(p'(x) = 2(x-a)q(x) + (x-a)^2 q'(x)\). Evaluate this at \(x=a\), leading to \(p'(a) = 0\).

Step 4: Evaluate p''(a)

Differentiate \(p'(x)\) to get \(p''(x) = 2q(x) + 4(x-a)q'(x) + (x-a)^2 q''(x)\). Evaluate at \(x=a\) gives \(p''(a) = 2q(a)\). Since \(q(a) eq 0\), \(p''(a) eq 0\). So, these conditions are satisfied.

Step 5: Converse - Derive p(x) from conditions

Assume \(p(a)=0\), \(p'(a)=0\), and \(p''(a) eq 0\). This suggests that \(p(x)\) can be expressed in the form \((x-a)^2 r(x)\). Differentiating and using the derivative conditions ensures \(r(x)\) is non-zero at \(a\). Thus, \(p(x)\) has a root of order 2.

Step 6: Conclusion

The proof shows that having \(p(a)=0\), \(p'(a)=0\), and \(p''(a) eq 0\) is both necessary and sufficient for \(p(x)\) to have a root of order 2 at \(x=a\). Therefore, the proposition is proved.

Key Concepts

Derivatives of Polynomials

Polynomials are expressions made up of variables and coefficients, using operations of addition, subtraction, multiplication, and non-negative integer exponents. In calculus, the derivative of a polynomial represents the rate at which the polynomial's value changes as the input changes. This process is known as differentiation.

Differentiation helps in understanding how polynomials behave and in finding important properties like slopes of tangents and identifying points of interest such as local maxima, minima, and roots. For a polynomial function \( p(x) \), its first derivative \( p'(x) \) is calculated by applying the power rule: For any term \( ax^n \), the derivative becomes \( anx^{n-1} \).

This first derivative informs us about the slope or the rate of change of the polynomial at any point \( x \). When necessary, higher-order derivatives, such as the second derivative \( p''(x) \), can be used to gain further insight into the polynomial's behavior, particularly in the context of root analysis and curve concavity.

Root Multiplicity

The concept of root multiplicity helps us understand how many times a particular value fulfills the root condition in a polynomial equation. If a polynomial \( p(x) \) has a root at \( x = a \), it means that \( p(a) = 0 \). But if the output of \( p(x) \) includes a factor like \( (x-a)^k \), where \( k > 1 \), then \( x = a \) is a root with multiplicity \( k \), often termed as a root of order \( k \).

In simpler terms, the multiplicity of a root indicates how many times that particular root repeats. For instance, a root of order 2 is also called a double root, while a root of order 3 is a triple root.

This idea is crucial in understanding polynomial equations' structure because it affects the behavior of the graph of the polynomial at \( x = a \). Such graphs might just 'bounce off' the x-axis at the repeated roots, which drastically changes how we interpret these polynomial functions.

Second Derivative Test

When examining polynomial roots of specific orders, the second derivative test becomes highly beneficial. It helps confirm the behavior and nature of a polynomial near a potential root. Typically, this test is used to determine whether a point is a local maximum, minimum, or a point of inflection, but it's also essential in verifying root orders.

For a polynomial \( p(x) \), if it is suspected to have a root of order 2 at \( x = a \), then:\

• \( p(a) = 0 \)
• \( p'(a) = 0 \)
• \( p''(a) eq 0 \)

These conditions ensure that the behavior around \( x = a \) fulfills the criteria of a double root. Here, the non-zero second derivative \( p''(a) eq 0 \) serves as a confirmation that the curve doesn't simply flatten out at the root but instead maintains some curvature. This helps in distinguishing between a true double root and other types of critical points.