Problem 5 - Algebra for College Students

Question

Perform the indicated divisions of polynomials by monomials. $$ \frac{15 a^{3}-25 a^{2}-40 a}{5 a} $$

Step-by-Step Solution

Answer

The simplified expression is \(3a^2 - 5a - 8\).

Step 1: Understand the Division Problem

The exercise requires us to divide a polynomial \(15a^3 - 25a^2 - 40a\) by the monomial \(5a\). The goal is to simplify this expression by performing the division.

Step 2: Divide Each Term Separately

Separate the polynomial into its individual terms so that each term can be divided by the monomial separately. The expression becomes \(\frac{15a^3}{5a} - \frac{25a^2}{5a} - \frac{40a}{5a}\).

Step 3: Perform Division for Each Term

Now, divide each term of the polynomial by the monomial: - \(\frac{15a^3}{5a} = 3a^{3-1} = 3a^2\) - \(\frac{25a^2}{5a} = 5a^{2-1} = 5a\) - \(\frac{40a}{5a} = 8a^{1-1} = 8\).

Step 4: Write the Final Simplified Expression

Compile the results of the divisions into the final expression: \(3a^2 - 5a - 8\).

Key Concepts

Understanding Monomials

A monomial is a term in algebra that represents a single term consisting of a constant, a variable, or a product of these, often raised to a power. For example, in the expression \( 5a \), "5" is the coefficient, and "\( a \)" is the variable. Here are some key characteristics of monomials:

• They contain only one term.
• The term can have constants, variables, and exponents (as long as they are non-negative integers).
• Examples include \(7x^2\), \(2a\), and simply \( 5 \) (a constant).

In problem-solving, monomials are often used in operations such as addition, subtraction, multiplication, and division with other algebraic expressions. They serve as a building block for polynomials, which are sums of several monomials.

Simplifying Expressions Through Polynomial Division

Simplifying expressions is a core part of efficiently handling algebraic problems. When diving into polynomial division, simplifying expressions means reducing complex polynomial terms by dividing them with simpler monomial terms. Here’s how to approach it:
Each term of the polynomial in the exercise is separately divided by the monomial. This simplifies the polynomial into a less complex expression. For example, when dividing \(\frac{15a^3}{5a}\), you simplify by dividing the coefficients (15 divided by 5 equals 3) and applying the law of exponents, which results in \(a^{3-1} = a^2\).
The same process applies to each term, treating each division independently, thus making it easier to manage. The main goal is to remove common factors and reduce the expression to its simplest form. This technique is crucial as it can greatly ease the solving of complicated algebraic equations.

Understanding Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations (addition, subtraction, multiplication, division). They can range from a simple expression like \(2x\) to complex ones involving multiple terms and operations. In our division example, \(15a^3 - 25a^2 - 40a\) is an algebraic expression consisting of three terms.

• Each term is a product of a number (coefficient) and a variable raised to an exponent.
• The terms are combined using addition or subtraction signs.

They are foundational in algebra, serving as a way to abstractly model and solve various mathematical problems. Simplifying such expressions by dividing each term by a common factor, like a monomial, is a powerful tool. It allows you to turn a complicated expression into a more manageable form and helps with further algebraic manipulation or solving equations.