Problem 28 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ 5 b-9 a b^{2}+10 a^{3} b-8 a b^{2}-9 a^{3} b $$

Step-by-Step Solution

Answer

Simplified: \( a^3b - 17ab^2 + 5b \)

Step 1: Group Like Terms

Begin by identifying terms where the variables have the same powers. In this expression, terms can be grouped as follows:- Terms involving \( b \) and \( a^{0} \): \(5b\)- Terms involving \( ab^2 \): \(-9ab^2\) and \(-8ab^2\)- Terms involving \( a^3b \): \(10a^3b\) and \(-9a^3b\).

Step 2: Simplify Each Group

Now, simplify each group of like terms:- Simplifying \( -9ab^2 - 8ab^2 \) gives: \(-17ab^2\).- Simplifying \( 10a^3b - 9a^3b \) gives: \(1a^3b\), which simplifies to \(a^3b\).

Step 3: Combine the Terms in Order

Now that each group has been simplified, arrange the expression in descending order of powers:- Highest degree term first is \( a^3b \).- Followed by the \( ab^2 \) term, \(-17ab^2\).- Finally, the constant term is \( 5b \).Therefore, the expression in descending order is: \[ a^3b - 17ab^2 + 5b \].

Key Concepts

Understanding Like Terms

When you're working with polynomial expressions, recognizing and grouping like terms is crucial for simplification.

• Like terms are terms within an expression that have the same variables raised to the same exponents. This means that both the variable and the exponent must be identical.
• For example, in the expression provided, the terms \(-9ab^2\) and \(-8ab^2\) are like terms because they both have the form \(ab^2\).
• Similarly, \(10a^3b\) and \(-9a^3b\) are like terms as they both involve \(a^3b\).

Recognizing these patterns is fundamental because it allows you to group them together and combine their coefficients, turning a complex expression into something more manageable. Be sure to not confuse terms that appear similar; always check that both the variables and their powers match exactly before considering them like terms.
This clarification ensures precision during the simplification process.

Arranging Polynomials in Descending Order

Writing a polynomial in descending order is often a requirement in mathematics to ensure consistency and clarity, especially when comparing polynomials.

• Descending order refers to arranging the terms of a polynomial expression starting with the highest power of a specific variable and moving down to the lowest.
• This structure allows for ease of interpretation as the degree of the terms clearly decreases.

For example, in our simplification exercise, we began with the term having the highest degree for the variable \(a\), which is \(a^3b\), and arranged lower degree terms \(-17ab^2\) and \(5b\) accordingly.
Consistency in arranging terms helps in communicating how the power of variables influences the overall value of the polynomial. It also facilitates further operations such as addition, subtraction, or even division of polynomials.

What is a Polynomial Expression?

A polynomial expression is a mathematical sentence involving a sum of powers in one or more variables multiplied by coefficients.

• This type of expression is distinguished by having whole number exponents and coefficients, with operations limited to addition, subtraction, and multiplication.
• For instance, the expression \(a^3b - 17ab^2 + 5b\) is a polynomial of multiple variables \(a\) and \(b\).
• The degree of a polynomial is determined by the highest sum of exponents within any single term. In this expression, the degree is 4 (from the term \(a^3b\)).

Polynomials form the foundation for more advanced topics in mathematics and their properties, such as roots and behaviors, are fundamental to understanding mathematical theorems and models. Mastering polynomial expressions helps in developing algebraic manipulation skills necessary for solving equations and modeling real-world problems.