Problem 15 - Elementary and Intermediate Algebra

Question

a. Write \(x-9+3 x^{2}+5 x^{3}\) in descending powers of \(x\) b. Write \(-2 x y+y^{2}+x^{2}\) in ascending powers of \(y\)

Step-by-Step Solution

Answer

a. \(5x^{3} + 3x^{2} + x - 9\); b. \(x^{2} - 2xy + y^{2}\)

Step 1: Identifying the powers of x

In the expression \(x-9+3x^{2}+5x^{3}\), identify the terms by their powers of \(x\): \(5x^{3}\), \(3x^{2}\), \(x\) (which is \(x^1\)), and the constant term \(-9\).

Step 2: Arranging terms in descending powers of x

Rearrange the terms identified in Step 1 so that they are in descending order of their \(x\) powers: \(5x^{3} + 3x^{2} + x - 9\). This expression now starts with the highest power of \(x\) and goes down to the lowest.

Step 3: Identifying the powers of y

In the expression \(-2xy + y^2 + x^2\), identify the terms based on the powers of \(y\): There are terms with \(y^1\), \(y^2\), and \(y^0\) (i.e., no \(y\)), which are \(-2xy, y^2, \) and \(x^2\) respectively.

Step 4: Arranging terms in ascending powers of y

Rearrange the terms identified in Step 3 so that they are in ascending order of their \(y\) powers: \(x^2 - 2xy + y^2\). This expression starts with the term having the lowest power of \(y\) and ends with the highest.

Key Concepts

Descending powers of x

When dealing with polynomial expressions, arranging terms by descending powers of a specific variable, such as \(x\), is a fundamental concept. To organize a polynomial in descending powers of \(x\), identify each term by how many times \(x\) is a factor in the term, also known as the term's degree. For example, consider the polynomial \(x - 9 + 3x^2 + 5x^3\). Here, the powers of \(x\) in each term are as follows:

• \(5x^3\) has \(x\) raised to the power of 3;
• \(3x^2\) has \(x\) to the power of 2;
• \(x\) is equivalent to \(x^1\);
• \(-9\) is a constant term with no \(x\), making its power of \(x\) zero.

To write this polynomial in descending powers of \(x\), rearrange the terms from the highest power to the lowest: \(5x^3 + 3x^2 + x - 9\). This method provides a clear view of the polynomial's structure, making subsequent algebraic manipulation easier.

Ascending powers of y

Arranging polynomial expressions by ascending powers of a variable is another critical skill. Focus on the expression \(-2xy + y^2 + x^2\). Here, the powers of \(y\) need to be rearranged in ascending order. Identifying these powers of \(y\), we see:

• \(x^2\) has a \(y\) power of 0 since \(y\) is absent;
• \(-2xy\) includes \(y^1\);
• \(y^2\) inherently has \(y\) to the power of 2.

Rearranging the terms from the lowest power of \(y\) to the highest yields \(x^2 - 2xy + y^2\). Organizing polynomials this way can provide clarity, particularly when dealing with functions involving multiple variables, helping to streamline calculations and solve mathematical problems efficiently.

Mathematical rearrangement

Rearranging terms in polynomial expressions is more than just a cosmetic change— it's an essential mathematical process for problem-solving and simplifying expressions. By organizing expressions like \(x - 9 + 3x^2 + 5x^3\), or \(-2xy + y^2 + x^2\), one creates a standardized format, enhancing readability and flexibility in mathematical operations.In practice, this involves identifying the degree of each term related to the selected variable— either \(x\), \(y\), or any other. Once identified, the terms are sorted either in descending or ascending order based on the requirements. This restructuring makes further mathematical manipulations, such as addition, subtraction, or even differentiation, much clearer and more manageable.Rearranging is crucial in solving equations involving polynomials as it helps recognize patterns and apply the appropriate algebraic rules. It is also widely used in polynomial division and factoring, laying a solid foundation for more advanced topics in algebra.

Polynomial terms

Polynomial terms form the building blocks of polynomial expressions. Each term consists of variables raised to an exponent with a coefficient in front and sometimes a constant term. For example, in the expression \(3x^2 - 2xy + 5\), each polynomial term can be identified as:

• \(3x^2\) - a term with \(x\) raised to the second power, multiplied by 3;
• \(-2xy\) - a term combining variables \(x\) and \(y\), both of degree 1;
• \(5\) - a constant term independent of any variables.

Understanding these components helps in categorizing and rearranging terms.Terms in a polynomial are significant because they determine the expression's degree, which is the highest power of the variable present. This degree helps in analyzing the polynomial's behavior, stability, and roots. A solid grasp of polynomial terms allows for easier factorization, solving equations, and graphing functions, which are necessary in more complex mathematical and real-world applications.