Problem 43 - Algebra 1

Question

Statement Explanation 1\. \(\frac{a c}{b c}=\frac{a}{b} \cdot \frac{?}{?}\) 1\. Apply the rule for multiplying rational expressions. 2\. \(\underline{?}=\frac{a}{b} \cdot \underline{?}\) 2\. Any nonzero number divided by itself is 1. 3\. \(\underline{?}=\frac{a}{b}\) 3\. Any nonzero number multiplied by 1 is itself. Use the method from Exercise 42 to show that \(\frac{2 x-4}{x^{2}-4}=\frac{2}{x+2}\)

Step-by-Step Solution

Answer

The given complex fraction \(\frac{2x-4}{x^{2}-4}\) simplifies to \(\frac{2}{x+2}\) after factoring and cancelling the common factors.

Step 1: Factoring the numerator and the denominator

Factor the numerator and denominator of the complex fraction. The numerator \(2x-4\) can be factored into \(2(x-2)\), and the denominator \(x^{2}-4\) can be factored by using the difference of squares method into \((x-2)(x+2)\). So it becomes \(\frac{2(x-2)}{(x-2)(x+2)}\)

Step 2: Cancel the common factor

Now, notice that \(x-2\) is both in the numerator and the denominator; they can be cancelled out. The equation thus becomes \(\frac{2}{(x+2)}\)

Step 3: Final Answer

After cancelling the common factors in the numerator and denominator, we are left with \(\frac{2}{x+2}\). This should be your final answer.

Key Concepts

Factoring Polynomials

When you come across a polynomial in mathematics, 'factoring' is essentially the process of breaking it down into simpler components that, when multiplied together, give you back the original polynomial. Imagine it as the mathematical equivalent of taking apart a Lego structure into the individual blocks. For example, when you factor the quadratic polynomial \(x^2 - 4\), it can be broken down into \((x - 2)(x + 2)\).

This process is used widely in simplifying rational expressions, solving equations, and many other algebraic operations. To factor a polynomial:

• Look for the greatest common factor among the terms.
• If it's a quadratic expression, try to express it as the product of two binomials.
• For more complex polynomials, look for patterns or use methods like grouping or synthetic division.

Factoring requires practice to identify the most efficient method quickly.

Difference of Squares

A special factoring technique comes into play when you encounter an expression that is a 'difference of squares'. This pattern occurs in polynomials where you have the subtraction of two squared terms, for example, \(a^2 - b^2\). It's termed a 'difference' because of the subtraction (minus sign) and 'of squares' since both terms are perfect squares.

The beauty of this expression is that it can always be factored into \((a + b)(a - b)\). Applying this knowledge can significantly simplify solving and understanding equations and rational expressions. In practice, you directly break down something like \(x^2 - 4\) into \((x - 2)(x + 2)\), slicing the problem in half, figuratively speaking.

Cancelling Common Factors

Cancelling common factors is like reducing a fraction to its simplest form. To cancel, you need a factor that appears both in the numerator (the top of a fraction) and the denominator (the bottom of a fraction). When you divide these common factors, they simplify to 1, based on the principle that any nonzero number divided by itself equals 1.

For instance, in the expression \(\frac{2(x-2)}{(x-2)(x+2)}\), the term \(x-2\) is present in both the numerator and the denominator. These can be cancelled out, leaving you with \(\frac{2}{x+2}\), which is a much simpler expression. Remember, this process only works if the factors are exactly the same and not just similar looking. Cancelling common factors not only helps with simplification but also prepares the expression for further mathematical operations.