Problem 17 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{21 a-9}{-3} $$

Step-by-Step Solution

Answer

Answer: -7a + 3

Step 1: Look at the given equation.

The given equation is $$ \frac{21 a - 9}{-3} $$ We need to divide the numerator by the constant in the denominator.

Step 2: Perform the divisions.

We have to divide each term of 21a -9 by -3. This gives us $$ \frac{21 a}{-3} - \frac{9}{-3} $$

Step 3: Simplify the divisions.

Now simplify both the divided terms: $$ -7a + 3 $$ So the division simplification of the given equation is $$ -7a + 3 $$

Key Concepts

Simplification

Simplification is the process of making an expression easier to work with or understand. When dividing algebraic expressions, simplification involves reducing the expression to its simplest form. This process generally includes performing arithmetic operations and canceling out similar terms. In the given division problem, simplifying was achieved by dividing each term of the expression \(21a - 9\) by the constant \(-3\). Each term in the numerator is treated as a separate mini-problem to be simplified.

• The term \( 21a \) was divided by \(-3\) which resulted in \(-7a\).
• The second term \(-9\) divided by \(-3\) resulted in \(+3\).

Together, these simplifications turned the complex expression \( \frac{21a - 9}{-3} \) into the simplified expression \(-7a + 3\). Simplification makes it far easier to handle this expression in further calculations.

Numerator and Denominator

Understanding the roles of the numerator and the denominator is crucial when dividing expressions. The numerator is the top part of a fraction and represents the quantity that is being divided. In our example, the numerator is \(21a - 9\). It is composed of two parts, the term with the variable \(21a\) and the constant \(-9\). Each part must be handled separately when dividing.The denominator is the bottom part, telling us the number by which the numerator is divided. Here, the denominator is \(-3\). It's important to remember when working with algebraic fractions:

• The numerator must be divided by the denominator, term by term.
• Always perform the operation carefully to avoid errors.

Understanding these roles will help ensure accuracy and fluency in operations involving fractions.

Polynomial Division

Polynomial division is a method used to divide polynomials just like numbers. In our problem, although technically a basic polynomial, the expression \(21a - 9\) has terms that need to be divided by a real number, \(-3\).Steps for polynomial division include:

• Separate the terms in the numerator and think of the denominator handing out to each of those terms separately.
• Perform division for each term individually, applying the arithmetic operations carefully. So for \(\frac{21a}{-3}\), this results in \(-7a\), and \(\frac{-9}{-3}\) results in \(+3\).
• Finally, recombine these individual results as the final simplified polynomial.

Here, dividing the polynomial by a constant simplifies the polynomial into a more manageable form \(-7a + 3\). The required goal of polynomial division is to break down and simplify, whether dealing with variables or numbers, enhancing your problem-solving toolbox in algebra.