Problem 30 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{16 a x^{2}-20 a x^{3}+24 a x^{4}}{6 a^{4}} $$

Step-by-Step Solution

Answer

Answer: The result of the division is \(\frac{8x^2}{3a^3} - \frac{10x^3}{3a^3} + \frac{4x^4}{a^3}\).

Step 1: Identify the terms of the polynomial and the constant term to be divided by

In this problem, we have a polynomial with three terms, and we need to divide it by a constant term: $$ \frac{16 a x^{2}-20 a x^{3}+24 a x^{4}}{6 a^{4}} $$ Polynomial terms: \(16ax^2, -20ax^3,\) and \(24ax^4\). Constant term: \(6a^4\).

Step 2: Divide each term of the polynomial by the constant term

Now we will divide each term of the polynomial by the constant term, simplifying the coefficients and canceling out common factors when possible: $$ \frac{16ax^2}{6a^4} - \frac{20ax^3}{6a^4} + \frac{24ax^4}{6a^4} $$

Step 3: Simplify each fraction and cancel common factors

We can simplify each fraction and cancel common factors as follows: $$ \frac{16}{6}\frac{a}{a^4}\frac{x^2}{1} - \frac{20}{6}\frac{a}{a^4}\frac{x^3}{1} + \frac{24}{6}\frac{a}{a^4}\frac{x^4}{1} \\ = \frac{8}{3}\frac{1}{a^3}x^2 - \frac{10}{3}\frac{1}{a^3}x^3 + \frac{4}{1}\frac{1}{a^3}x^4 $$

Step 4: Write the final simplified expression

Now, let's write the final simplified expression by combining the fraction and the remaining terms: $$ \frac{8x^2}{3a^3} - \frac{10x^3}{3a^3} + \frac{4x^4}{a^3} $$ So, the final result of the division is: $$ \frac{16 a x^{2}-20 a x^{3}+24 a x^{4}}{6 a^{4}} = \frac{8x^2}{3a^3} - \frac{10x^3}{3a^3} + \frac{4x^4}{a^3} $$

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental concept in algebra used to make complicated expressions easier to work with. In this context, simplifying involves reducing expressions to their simplest form. Let’s break this down with an example. Suppose you have an expression like \(16ax^2 - 20ax^3 + 24ax^4\). This expression is initially complex. By dividing each term by a common factor, such as \(6a^4\), and simplifying each result, we make the expression shorter and easier to interpret.

Here are some steps to simplify:

• Identify common factors in each term.
• Divide each term by these factors to reduce the terms.
• Simplify the coefficients as much as possible (e.g., reducing fractions).

In simplifying, you might also combine like terms, but our example focuses on expression terms that are simplified individually. After simplification, we arrive at a more straightforward version, such as \(\frac{8x^2}{3a^3} - \frac{10x^3}{3a^3} + \frac{4x^4}{a^3}\).

Simplification is an essential skill that plays a role not just here, but in any mathematical problem solving. It helps to see relationships and patterns more easily and ensures you work with manageable numbers.

Fraction Simplification

Fraction simplification occurs when you take a fraction and reduce it to its simplest form. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).

In the given example:

• \(\frac{16}{6}\), \(\frac{20}{6}\), and \(\frac{24}{6}\) all have common divisors that can simplify the fractions.
• To simplify \(\frac{16}{6}\), you find both numbers share the number 2 as a factor, leading to \(\frac{8}{3}\).
• If different numerical terms shared larger GCDs, like a factor of 4 or more, you would leverage those for more simplification.

This process eases the fraction representation by making it less cumbersome.

When simplifying algebraic fractions, you also consider algebraic terms much like the coefficients. For instance, \(\frac{a}{a^4}\) simplifies to \(\frac{1}{a^3}\) by cancelling out common powers of \(a\). By simplifying, you avoid working with needlessly large or unwieldy numbers, making calculations and comparisons easier.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator are algebraic expressions. Just like numeric fractions, algebraic fractions require simplification for ease of computation.

Our example, \(\frac{16ax^2 - 20ax^3 + 24ax^4}{6a^4}\), is an algebraic fraction because it contains variables, \(a\) and \(x\). To simplify algebraic fractions, follow these steps:

• Identify common algebraic factors in the numerator and the denominator.
• Reduce these factors just like you would with numbers. This involves cancelling variables when possible (e.g., reducing \(a/a^4\) to \(1/a^3\)).
• Handle each term individually if the polynomial is more complex.

This simplification allows one to easily analyze and compute the given expression. The result after simplification is easy to interpret and use. With the given example, you break down the expression to \(\frac{8x^2}{3a^3} - \frac{10x^3}{3a^3} + \frac{4x^4}{a^3}\).

Working with algebraic fractions effectively reduces potential errors during further computations, saving time and effort.