Problem 57 - Calculus Single Variable

Question

Suppose that \(p(x)\) is a nonzero polynomial. Prove that \(p(x)\) has degree precisely \(k\) if and only if a. \(p^{(k+1)}(x)=0\) for all \(x\); and b. \(p^{(k)}\left(x_{0}\right) \neq 0\) for some \(x_{0}\).

Step-by-Step Solution

Answer

A polynomial degree \(k\) is confirmed by \(p^{(k+1)}(x)=0\) for all \(x\) and \(p^{(k)}(x_0) \neq 0\) for some \(x_0\).

Step 1: Understanding Polynomial Degree

A polynomial \(p(x)\) of degree \(k\) is generally written as \(p(x) = a_k x^k + a_{k-1} x^{k-1} + \ldots + a_0\), where \(a_k\) is non-zero.

Step 2: Calculating Higher-Order Derivatives

The derivative reduces the degree of a polynomial by 1. The \((k+1)\)-th derivative of \(p(x)\) will nullify all terms of degree **equal to or less** than \(k\) to 0, hence \(p^{(k+1)}(x) = 0\) for all \(x\). This confirms part 'a', showing that the degree of \(p(x)\) cannot be greater than \(k\).

Step 3: Verification of Degree \(k\)

To show that degree is at least \(k\), the \(k\)-th derivative \(p^{(k)}(x)\) will differentiate the leading term \(a_k x^k\) into \(k! \cdot a_k\), which is a non-zero constant, provided \(a_k eq 0\). Thus, for some \(x_0\), \(p^{(k)}(x_0) eq 0\) confirms this degree.

Key Concepts

Higher-Order Derivatives

Derivatives are a powerful tool in calculus that help to analyze the behavior of functions. When we talk about the **derivatives** beyond the first one, these are known as **higher-order derivatives**. Simply put, each derivative you take of a polynomial will decrease its degree by one. Let's understand this more:

• The first derivative of a polynomial function reduces its degree by one. If the original degree is **k**, the first derivative will be **k-1**.
• Continuing in this manner, the second derivative will have a degree of **k-2** and so on.
• Therefore, by the time you reach the **(k+1)**-th derivative, all terms of the polynomial have been differentiated away, resulting in the zero polynomial: \[ p^{(k+1)}(x) = 0 \]

Understanding higher-order derivatives provides insight into the maximum degree the original polynomial can have. If the degree is indeed **k**, then any derivative beyond **k** will be zero, confirming the polynomial's degree. This concept is crucial in identifying and verifying the degree of polynomial functions.

Differentiation

Differentiation is the process of calculating a derivative, which gives us an important way to analyze changes within functions, specifically polynomials here. For a polynomial function, differentiation helps us to:

• Determine the rate of change or slope of the polynomial function at any point within its range.
• Simplify the process of identifying the behavior of the polynomial through different orders of derivatives.

For a polynomial of degree **k**, differentiation plays a key role in revealing its internal structure. As mentioned, the differentiation step reduces the polynomial's degree systematically. Through several steps of derivatives, each coefficient's significance becomes evident until only the leading term's coefficient is active in the **k-th** derivative. This systematic approach imbues the differentiation process with the power to validate not only the function's behavior but also its **degree**, an important aspect to confirm its characteristics.

Verification of Degree

Verification of a polynomial's degree ensures that we have correctly identified the highest power within the function. In this case, it involves a clever use of differentiation:

• By differentiating a polynomial **k** times, a polynomial can be verified as having degree **k** if \[ p^{(k)}(x_0) eq 0 \] for at least one point \( x_0 \).
• This ensures that we are left with the leading coefficient multiplied by **k!**, a non-zero value if the actual degree of the polynomial is indeed **k**.
• Therefore, reaching a point where the \((k+1)\)-th derivative is zero confirms there are no terms of higher degree.

This process of verification not only solidifies understanding but also proves one's analytical computations about the polynomial, ensuring its proper degree designation.