Question

\(p(x)=3-x^4\)

Short answer

This exercise asks us to analyze the polynomial \(p(x) = 3 - x^4\). It is a 4th-degree polynomial that is continuous and differentiable everywhere in its domain. It cannot be simplified any further.

Step-by-step solution

Identify the Polynomial

The function given is \(p(x) = 3 - x^4\), which is a polynomial. The term \(3\) is a constant, while the term \(-x^4\) is a degree-4 monomial.

Simplify the Polynomial

In this case, the polynomial is already in its simplest form and cannot be simplified any further.

Analyzing the polynomial

Without a specific task, there are several things we can state about this function. The function is a simple polynomial function of degree 4. Consequently, it is continuous and differentiable everywhere on the real number line.

Key concepts

Polynomial Degree

The degree of a polynomial is a crucial concept in understanding polynomial functions. A polynomial's degree is essentially the highest power of the variable in the polynomial expression. For example, consider the polynomial function given in the exercise: \[p(x) = 3 - x^4\]Here, the degree of the polynomial is 4, because the highest exponent of the variable \(x\) is 4. This means that if you were to graph this polynomial, it would potentially intersect the x-axis up to four times.
The degree of a polynomial gives us important information about its behavior, such as the number of solutions (roots) it can have and how it behaves as \(x\) approaches infinity. In the case of a degree 4 polynomial like this one, the graph can exhibit up to one less than the degree (i.e., 3) hills and valleys.

Monomials

A monomial is a single term in a polynomial, consisting of a constant multiplied by a variable raised to a non-negative integer power. In the polynomial described by \[p(x) = 3 - x^4\]we find that it comprises two terms: the constant term \(3\) and the monomial \(-x^4\).
Monomials form the building blocks of polynomials. They are important because understanding each monomial helps us understand the polynomial's degree and its behavior. For example:

• The degree of the polynomial is determined by the highest degree of its monomials.
• A negative sign in front of a monomial, like \(-x^4\), reflects across the x-axis when graphed.

The simplicity of the monomial term makes it easier to analyze and predict the overall shape and direction of the polynomial function.

Continuity

Continuity is a key feature of polynomial functions, and it ensures that the graph of a function can be drawn without lifting the pencil from the paper. In intuitive terms, a continuous function is one that doesn’t have any breaks, jumps, or holes. All polynomial functions, including the one given, are continuous everywhere on the real number line.For the polynomial \(p(x) = 3 - x^4\), continuity means:

• The limits exist and are the same from both directions at every point on the curve.
• The function is smooth, with no sharp turns or corners.

Therefore, you can expect that the graph of this polynomial will be a smooth curve, seamlessly extending without interruption in both directions.

Differentiability

Differentiability goes hand in hand with continuity in many respects but adds yet another layer of detail. Whereas continuity ensures there are no jumps or breaks, differentiability ensures the function doesn't have sharp points where its slope becomes undefined.Given the polynomial \(p(x) = 3 - x^4\), you can rest assured that this function is differentiable over the entire real number line, meaning:

• You can find a derivative at any point, reflecting the slope of the tangent to the curve at that point.
• Its smooth nature implies that it is smooth and curved.

The derivative of this polynomial is \(p'(x) = -4x^3\), which is defined everywhere. Thus, if you need to find rates of change or the tangent line at any point, differentiability provides a straightforward pathway to these answers.