Problem 10 - Algebra for College Students

Question

Perform the indicated divisions of polynomials by monomials. $$ \frac{-27 a^{3} b^{4}-36 a^{2} b^{3}+72 a^{2} b^{5}}{9 a^{2} b^{2}} $$

Step-by-Step Solution

Answer

Result: \(-3 a b^{2} - 4 b + 8 b^{3}\).

Step 1: Understand the Problem

You are asked to divide a polynomial by a monomial. The polynomial in question is \[-27 a^{3} b^{4} - 36 a^{2} b^{3} + 72 a^{2} b^{5}\] and the monomial is \(9 a^{2} b^{2}\). We proceed by dividing each term of the polynomial by this monomial separately.

Step 2: Divide Each Term

For each term of the polynomial, we divide the term by the monomial. First term: \[-27 a^{3} b^{4} \div 9 a^{2} b^{2} = -3 a^{1} b^{2}\]Second term: \[-36 a^{2} b^{3} \div 9 a^{2} b^{2} = -4 b^{1}\]Third term: \[72 a^{2} b^{5} \div 9 a^{2} b^{2} = 8 b^{3}\]

Step 3: Simplify and Combine Results

Combine the simplified results for each term obtained in Step 2, giving the final result of the division as \[-3 a b^{2} - 4 b + 8 b^{3}\].

Key Concepts

Understanding Monomials

A monomial is a single, indivisible algebraic expression. It consists of a product of numbers and variables raised to whole number powers. For example, in our exercise, the monomial is \(9a^2b^2\). This monomial consists of a numerical coefficient, 9, and two variables, \(a\) and \(b\), each raised to an exponent of 2. Monomials are crucial building blocks in algebra, serving as the simplest algebraic expressions. You can think of them as basic "words" in the language of algebra.

Analyzing Polynomial Terms

In algebra, a polynomial is a sum of terms, where each term is a monomial. In our given problem, the polynomial is composed of three terms:

• \(-27a^3b^4\)
• \(-36a^2b^3\)
• \(72a^2b^5\)

Each term consists of a coefficient (e.g., -27, -36, 72) and variables raised to powers. The terms are separated by addition or subtraction signs. When we work with polynomials, each term can be handled individually, especially in operations like addition, subtraction, and division.

Mastering Simplification in Polynomial Division

Simplification is about reducing expressions to their simplest form. When dividing a polynomial by a monomial, simplification involves dividing each term of the polynomial by the monomial independently. For instance, divide \(-27a^3b^4\) by \(9a^2b^2\), and you simplify it to \(-3ab^2\). You reduce coefficients by dividing them, and reduce the powers of like variables by subtracting exponents. It’s essentially breaking down the complex expression bit by bit, making it manageable and easier to understand.

Exploring Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication) that represent a specific set of values or relationships. The expression \(-27a^3b^4 - 36a^2b^3 + 72a^2b^5\) combines terms into a complete polynomial.

• The structure of an algebraic expression directly influences how we handle operations like division or simplification.
• Variables can represent different numbers, their exponents showing how many times they are used in a multiplication.
• Coefficients are critical in defining the impact of each term in the expression.

Understanding these elements helps in manipulating and simplifying expressions and in performing operations like polynomial division effectively.