Problem 64 - Beginning and Intermediate Algebra

Question

Write a fourth-degree polynomial in \(x\) that does not contain a second-degree term.

Step-by-Step Solution

Answer

A possible fourth-degree polynomial in \(x\) that does not contain a second-degree term is: \(P(x) = x^4 + 2x^3 + 3x + 4\).

Step 1: Define the general form of a fourth-degree polynomial

The general form of a fourth-degree polynomial is: \(P(x) = ax^4 + bx^3 + cx^2 + dx + e\) Where a, b, c, d, and e are constants and \(a \neq 0\).

Step 2: Remove the second-degree term

Since the polynomial should not have a second-degree term, the coefficient of the \(x^2\) term, i.e., c should be equal to 0. So the polynomial becomes: \(P(x) = ax^4 + bx^3 + dx + e\) Now we can choose any values for a, b, d, and e to create a specific fourth-degree polynomial without a second-degree term.

Step 3: Choose the coefficients

Let's choose some values for the coefficients a, b, d, and e: a = 1, b = 2, d = 3, e = 4

Step 4: Write down the final polynomial

Substituting the chosen coefficient values in \(P(x)\), we get: \(P(x) = 1x^4 + 2x^3 + 3x + 4\) So, a possible fourth-degree polynomial in \(x\) that does not contain a second-degree term is: \(P(x) = x^4 + 2x^3 + 3x + 4\)

Key Concepts

Understanding Polynomial Expressions

A polynomial expression is a mathematical expression that involves a sum of powers in one or more variables. In this case, we are working with a single variable, \(x\). Each term can have:

• a numerical coefficient,
• powers of the variable, and
• a constant term.

Polynomials of different degrees have different maximum powers. For example, a fourth-degree polynomial means the highest power of \(x\) is 4. The general form of a fourth-degree polynomial is \( ax^4 + bx^3 + cx^2 + dx + e \), where \(a\), \(b\), \(c\), \(d\), and \(e\) are constants, and \( a eq 0 \).
In some cases, you may need to write a polynomial without specific terms, like in our example where we exclude the second-degree term. Understanding how to modify the polynomial expression will help you create polynomials fitting specified needs.

Exploring Algebraic Terms in Polynomials

Algebraic terms are the building blocks of polynomials. Each term is composed of a coefficient multiplied by the variable \(x\) raised to a certain power. For our fourth-degree polynomial:

• \( ax^4 \) is the fourth-degree term,
• \( bx^3 \) is the third-degree term,
• \( cx^2 \) is the second-degree term,
• \( dx \) is the first-degree term,
• \( e \) is the constant term.

Each of these terms contributes differently to the behavior of the polynomial expression. The degree of the polynomial is determined by the highest power of \(x\) present, which in this case is a power of 4, due to the \( ax^4 \) term. When asked to exclude a certain term, such as the second-degree term in this case, we set the coefficient \(c\) to zero, effectively removing that term from the expression.

Coefficient Selection and its Importance

Coefficients are the numerical factors in terms of a polynomial expression, and their selection is crucial as it determines the shape and position of the curve that the polynomial represents when graphed.
For a given polynomial, different choices of coefficients will lead to different functions.Here is a breakdown of the coefficient selection process for our example:

• Since it's a fourth-degree polynomial, \(a\) cannot be zero, or else it won't be a fourth-degree expression.
• For this exercise, we chose specific coefficients: \(a = 1\), \(b = 2\), \(d = 3\), and \(e = 4\).
• The choice of these coefficients gives us a particular function \(P(x) = x^4 + 2x^3 + 3x + 4\).

To not include a term, simply set its coefficient to zero. Here, \( c = 0 \) to exclude \(x^2\). Choosing coefficients appropriately can make the polynomial satisfy particular conditions or constraints needed for a problem or model.