Problem 29 - Algebra and Trigonometry

Question

Perform the multiplication or division and simplify. $$ \frac{4 x}{x^{2}-4} \cdot \frac{x+2}{16 x} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{4(x-2)}\).

Step 1: Factor the Denominator

Identify and factor the quadratic expression in the denominator of the first fraction. The expression is \(x^2 - 4\), which is a difference of squares. It can be factored as \((x-2)(x+2)\).

Step 2: Rewrite the Expression

Rewrite the entire expression using the factored form of the denominator: \(\frac{4x}{(x-2)(x+2)} \cdot \frac{x+2}{16x}\).

Step 3: Cancel Common Factors

Identify and cancel the common factors in the numerators and denominators. The factor \(x+2\) appears in both the numerator of the second fraction and the denominator of the first fraction, and the factor \(x\) appears in both the numerator of the first fraction and the denominator of the second fraction. Cancel these common factors.

Step 4: Multiply Remaining Expressions

After cancelling, you are left with \(\frac{4}{(x-2) \, 16}\). Multiply the numbers and expressions that remain: \(\frac{4}{16} = \frac{1}{4}\). The denominator also has \((x - 2)\), making the final expression \(\frac{1}{4(x-2)}\).

Step 5: Write the Final Simplified Expression

The simplified result, after performing the multiplication and cancelling the common factors, is \(\frac{1}{4(x-2)}\).

Key Concepts

Rational Expressions

Rational expressions are similar to fractions, but instead of just numbers in the numerator and denominator, they can contain polynomials. Understanding rational expressions is crucial because they frequently appear in algebra.
They consist of a ratio formed by dividing two polynomials.

• Numerator: The top part of the fraction, often containing a polynomial.
• Denominator: The bottom part, which cannot be zero.

Simplifying rational expressions involves writing them in their simplest form by reducing the fraction. To do this, we:

• Factor both the numerator and the denominator.
• Cancel out any common factors.

Knowing how to simplify these expressions helps solve complex algebra problems more easily.

Factoring Polynomials

Factoring polynomials is a fundamental skill needed to solve many algebraic problems. It involves expressing a polynomial as a product of simpler polynomials.
Polynomials like the quadratic expression in our example, such as \(x^2 - 4\), can often be factored using specific methods. This one is a difference of squares because \(x^2-4 \) can be rewritten as \((x+2)(x-2)\).

• Common Methods: Look for common terms or use special formulas like difference of squares.
• Prime Polynomials: Some polynomials can't be factored further and are called prime polynomials.

Mastering this technique is essential for simplifying rational expressions as it allows us to cancel common terms.

Multiplying Fractions

Multiplying fractions, including those with polynomials, involves straightforward steps. When you multiply two fractions, you multiply the numerators and then the denominators.
For the expression \(\frac{4x}{(x-2)(x+2)} \cdot \frac{x+2}{16x}\), this means:

• Numerator: Multiply the terms in the numerators of the fractions together.
• Denominator: Do the same for the denominators.

Sometimes, however, you can simplify the fractions first by canceling common factors, making the multiplication easier.
Remember, it’s often beneficial to simplify fractions before multiplying as it results in smaller numbers, which are easier to work with.

Simplifying Expressions

Simplifying expressions involves combining like terms and performing operations to write expressions in their simplest form.
In our example, simplification meant identifying and canceling common factors between the numerator and the denominator of the multiple fractions.

• Cancel Factors: Look for terms that appear in both the numerator and the denominator and cancel them.
• Combine: Make sure to fully multiply and combine leftover terms.

Once simplified, the expression is in its most manageable form, such as \(\frac{1}{4(x-2)}\) in the example above. This makes it easier to understand and use in further calculations or problem solving.