Problem 18 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{3 x^{2}-6 x}{-3} $$

Step-by-Step Solution

Answer

Question: Perform the division and simplify the expression: $$\frac{3x^2 - 6x}{-3}$$ Answer: $$-1x^2 + 2x$$

Step 1: Analyze the expression

We have the expression: $$ \frac{3x^2 - 6x}{-3} $$ Our goal is to divide both terms in the numerator by the denominator and then simplify the expression.

Step 2: Divide the terms

We will divide each term in the numerator by the denominator: $$ \frac{3x^2}{-3} - \frac{6x}{-3} $$

Step 3: Simplify each term

Now, we will simplify each term obtained in the previous step: $$ (-1x^2) + (2x) $$

Step 4: Final simplified expression

After dividing and simplifying the terms, we get the final simplified expression: $$ -1x^2 + 2x $$

Key Concepts

Simplification

In mathematics, simplification is the process of reducing an expression to its simplest form. When performing simplification during polynomial division, the goal is to make the expression easier to work with, usually by reducing unnecessary parts and combining like terms. This process often involves recognizing patterns, canceling out common factors, and rearranging the terms to achieve a more concise expression.

In the exercise given, the expression \( \frac{3x^2 - 6x}{-3} \) was simplified by dividing each term in the numerator by the denominator. First, the expression was separated into two parts: \( \frac{3x^2}{-3} \) and \( \frac{6x}{-3} \). Then each term was individually simplified by performing the division, leading to \(-1x^2 + 2x\). Managing the negative signs and understanding how each term in the numerator interacts with the denominator is crucial during simplification to ensure results are accurate.

Numerator and Denominator

The terms *numerator* and *denominator* are essential in describing fractions. In polynomial division, understanding these terms is critical because the numerator is the expression on top of the fraction line, while the denominator sits on the bottom.

For the fraction \( \frac{3x^2 - 6x}{-3} \), the numerator is \(3x^2 - 6x\), consisting of multiple terms, and the denominator is \(-3\), a monomial. This means the expression requires us to divide each term of the numerator separately by the single term (denominator).

• The **numerator**: This tells us what's being divided. Here, it's a polynomial with terms \(3x^2\) and \(-6x\).
• The **denominator**: This informs us dividing is done with \(-3\) for each term in the numerator.

Understanding these roles allows you to apply the operation correctly, ensuring each division step treats the numerator and denominator appropriately.

Monomial Division

Monomial division involves dividing a polynomial by a monomial. This requires you to handle each term in the polynomial (numerator) separately with the monomial (denominator).

In the example \( \frac{3x^2 - 6x}{-3} \), we observe monomial division as we divide the two terms in the polynomial: \(3x^2\) and \(-6x\). Each is handled independently, dividing by \(-3\). When dividing, it helps to:

• Divide the coefficients (numbers in front of the variables) for each term separately.
• Adjust the exponents if any variable terms are involved, though in this specific case, only the coefficients needed adjustment.
• Carefully manage any negative signs during the division process.

These steps ensure the polynomial is correctly simplified, making further mathematical operations straightforward. Practicing with different problems can help build familiarity and ease with these procedures.