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Chapter 1: Problem 13

Perform the indicated divisions. $$\frac{3 a^{2} x+6 x y}{3 x}$$

Short Answer

Expert verified
The simplified expression is \(a^2 + 2y\).

Step by step solution

01

Simplify the Numerator

The numerator of the fraction is \(3a^2x + 6xy\). This expression can be factored by extracting the common factor, which is \(3x\). Thus, \(3a^2x + 6xy = 3x(a^2 + 2y)\).
02

Divide by the Denominator

The whole expression then becomes \(\frac{3x(a^2 + 2y)}{3x}\). Since \(3x\) is a common factor in both the numerator and the denominator, they can be cancelled out. The division simplifies to \(a^2 + 2y\).
03

Write the Final Simplified Expression

After cancelling the common factor \(3x\), the simplified expression is simply \(a^2 + 2y\). This is the result of the division.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Expressions
Factoring an expression involves writing it as a product of simpler terms or factors. In the exercise provided, the expression in the numerator is \(3a^2x + 6xy\). Our goal in factoring is to identify a common factor that can be extracted from each term.
  • First, observe that the numerator consists of two terms: \(3a^2x\) and \(6xy\).
  • Both terms have a common factor of \(3x\). This means \(3x\) can be factored out of the expression.
  • Factoring \(3x\) out of \(3a^2x + 6xy\) leads us to \(3x(a^2 + 2y)\).
Factoring is a crucial skill in algebra as it simplifies expressions, making them easier to manipulate or solve for variables.
Simplifying Fractions
Simplifying fractions is a key step in solving algebraic expressions, especially involving division. To simplify a fraction, you need to find any common factors between the numerator and the denominator and then cancel them out.
  • Once the numerator \(3a^2x + 6xy\) is factored as \(3x(a^2 + 2y)\), we compare it to the denominator \(3x\).
  • Since \(3x\) is present in both the numerator and the denominator, they can be cancelled, effectively simplifying the expression.
  • This step leaves us with \(a^2 + 2y\) as the final simplified expression.
Simplifying fractions is powerful as it reduces complex expressions into simpler forms, facilitating easier comprehension and application in subsequent calculations.
Polynomial Expressions
Polynomial expressions are fundamental in algebra and can take many forms, including terms with variables raised to exponents. In the exercise, the final simplified result \(a^2 + 2y\) is a polynomial expression.
  • It consists of two terms: \(a^2\) and \(2y\).
  • \(a^2\) is a term where the variable \(a\) is raised to the power of 2, showcasing the characteristic of polynomials having exponents.
  • Polynomials can often be categorized by their degree, which is determined by the highest exponent. In this expression, \(a^2\) shows that the degree is 2.
Understanding polynomial expressions is vital, as these forms are commonly used in higher-level algebra and calculus, influencing a wide range of mathematical and real-world applications.

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