Problem 4 - College Algebra

Question

The algebraic expression a polynomial? If it is, write the polynomial in standard form. $$x^{2}-x^{3}+x^{4}-5$$

Step-by-Step Solution

Answer

\(x^{4}-x^{3}+x^{2}-5\)

Step 1: Identify the Polynomial

The given expression, \(x^{2}-x^{3}+x^{4}-5\), is indeed a polynomial. It consists of variables and coefficients and the variables only have non-negative integer exponents

Step 2: Write Polynomial in Standard Form

To write this polynomial in standard form, we rearrange the terms so that the powers of \(x\) decrease from left to right. Hence, the standard form of the given polynomial becomes \(x^{4}-x^{3}+x^{2}-5\).

Key Concepts

Standard Form of a Polynomial

In algebra, writing a polynomial in its standard form is crucial for clarity and ease of understanding. The standard form of a polynomial means arranging the terms in descending order of their degrees. Each term of the polynomial is expressed with the highest power first and gradually reduces as you move across the terms.

For example, in the polynomial expression given:

• Original: \(x^{2}-x^{3}+x^{4}-5\)
• Standard Form: \(x^{4}-x^{3}+x^{2}-5\)

To achieve this, start by identifying the variable with the highest exponent. In this case, \(x^{4}\) has the largest exponent, so it becomes the leading term. Followed by the terms \(x^{3}\), \(x^{2}\), and finally, any constants come at the end. This method ensures consistency and a better understanding of polynomials for anyone examining them. Writing in standard form is particularly helpful for polynomial operations such as addition, subtraction, and comparison.

Non-negative Integer Exponents

Polynomials consist of terms that include variables raised to whole number exponents, which are the non-negative integers. These exponents are always zero or positive numbers.

• A polynomial can't have negative exponents.
• Non-negative exponents indicate the smooth and continuous nature of the polynomial graph.
• In our polynomial, the exponents are 2, 3, and 4, which are non-negative integers.

This characteristic differentiates polynomials from other expressions that might include fractions, square roots, or negative exponents. The presence of only non-negative integer exponents makes the mathematical manipulation of polynomials straightforward and predictable.

Variables and Coefficients in Polynomials

A polynomial is composed of terms, each containing variables and coefficients. Understanding these elements is essential for interpreting and constructing polynomials.

• Variables: Represent unknown values which are often denoted by letters such as \(x\), \(y\), etc. In our example, \(x\) is the variable used.
• Coefficients: are the numerical factors that multiply the variables. For instance, in \(2x^2\), 2 is the coefficient of the term.
• Each term in the polynomial is a product of a coefficient and the variable raised to a power, like \(x^2\). In some cases, like \(-5\), the coefficient stands alone without a variable.

Identifying and working with variables and coefficients skillfully allows for easy manipulation, simplifying, and solving of polynomials across various algebraic applications.