Problem 24 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{-12 x^{2}(x+4)}{4 x} $$

Step-by-Step Solution

Answer

Question: Reduce the given rational expression to its lowest terms: $$\frac{-12 x^2 (x+4)}{4x}$$ Answer: The simplified expression is \(-3x(x+4)\).

Step 1: Rewrite the expression

First, let's rewrite the given expression as: $$ \frac{-12 x^2 (x+4)}{4x} $$

Step 2: Factor out the common factors

In the numerator, we can factor out \(4x\) from \(12x^2\). This gives us: $$ \frac{-4x(3x)(x+4)}{4x} $$

Step 3: Cancel out the common factors

Now, we can cancel out the common factors of \(4x\) in the numerator and denominator: $$ \frac{-\cancel{4x}(3x)(x+4)}{\cancel{4x}} = -3x(x+4) $$

Step 4: Write the simplified expression

Finally, we'll write the simplified rational expression as: $$ -3x(x+4) $$ So, the given expression reduced to lowest terms is \(-3x(x+4)\).

Key Concepts

Factoring

Factoring is an essential skill in algebra that helps simplify complex expressions. It involves breaking down expressions into products of simpler expressions or numbers. In the given exercise, factoring is used to break down the numerator of a rational expression.

To factor a polynomial, you need to identify common factors in its terms. For instance:

• Look at each term and find the greatest common factor (GCF).
• In the exercise, the terms in the numerator, \(12x^2(x+4)\), were factored by recognizing \(4x\) as the GCF.
• Express the original terms as the product of their GCF and another term, which results here in \(-4x(3x)(x+4)\).

Factoring significantly reduces the complexity of expressions, leading to easier simplification. With practice, identifying factors becomes an intuitive process, empowering you to tackle more challenging algebraic tasks.

Simplifying Expressions

Simplifying expressions is about reducing them to their most compact form while maintaining the same value. This can make expressions more understandable and easier to work with.

In our exercise, the expression was simplified by:

• Finding common factors in both the numerator and the denominator, which was \(4x\) in this case.
• Once you identify these, you "cancel" them out to reduce the expression. Cancelling common factors means removing them from both the numerator and the denominator.
• After canceling, the expression is written in a simpler form, \(-3x(x+4)\).

Through simplifying expressions, equations and operations become less cumbersome and more straightforward, making subsequent calculations faster and more efficient. Simplification is particularly useful in solving equations and performing algebraic operations.

Algebraic Fractions

Algebraic fractions are like regular fractions, but they contain polynomials in the numerator, the denominator, or both. Working with algebraic fractions involves skills like factoring and simplifying.

Key points to consider when working with algebraic fractions include:

• Treat them just like numerical fractions - focus on the relationship between the numerator and the denominator.
• Use factoring to find common terms in both numerator and denominator.
• In the exercise, the algebraic fraction \(\frac{-12 x^{2}(x+4)}{4 x}\) requires recognizing common factors to simplify.

Once simplified, algebraic fractions are easier to evaluate or involve in further operations like addition, subtraction, multiplication, or division. Mastery of these takes practice but is invaluable in handling more advanced algebraic concepts efficiently.