Problem 42 - Algebra 2

Question

Which expression can be simplified to \(\frac{x-1}{x-3} ?\) $$ \begin{array}{ll}{\text { A. } \frac{x^{2}-x-6}{x^{2}-x-2}} & {\text { B. } \frac{x^{2}-2 x+1}{x^{2}+2 x-3}} \\ {\text { C. } \frac{x^{2}-3 x-4}{x^{2}-7 x+12}} & {\text { D. } \frac{x^{2}-4 x+3}{x^{2}-6 x+9}}\end{array} $$

Step-by-Step Solution

Answer

Expression C: \(\frac{x^{2}-3 x-4}{x^{2}-7 x+12}\) simplifies to \(\frac{x-1}{x-3}\). So, the correct answer is C.

Step 1: Simplify the expression A.

Rewrite the given expression by factoring the polynomials. On simplifying \(\frac{x^{2}-x-6}{x^{2}-x-2}\), we get \(\frac{(x-3)(x+2)}{(x-2)(x+1)}\). The simplified form doesn't match \(\frac{x-1}{x-3}\), so it's not the answer.

Step 2: Simplify the expression B.

Now, simplify \(\frac{x^{2}-2 x+1}{x^{2}+2 x-3}\). It simplifies to \(\frac{(x-1)^{2}}{(x+1)(x-3)}\). It doesn't simplify to \(\frac{x-1}{x-3}\), so this also doesn't give the correct answer.

Step 3: Simplify the expression C.

Now, let's simplify \(\frac{x^{2}-3 x-4}{x^{2}-7 x+12}\). This simplifies to \(\frac{(x-4)(x+1)}{(x-3)(x-4)}\). By cancelling out (x-4) from both the numerator and the denominator, we get \(\frac{x-1}{x-3}\). This matches perfectly with our required expression.

Step 4: Verification with expression D.

Though we've found our answer, for completeness, let's simplify expression D as well. Simplifying \(\frac{x^{2}-4 x+3}{x^{2}-6 x+9}\) gives us \(\frac{(x-1)(x-3)}{(x-3)^{2}}\). Hence, this is not equal to \(\frac{x-1}{x-3}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra, especially when working with rational expressions. A polynomial can be factored by finding its roots or zeros, essentially breaking it down into the product of its simpler polynomials. For instance, consider the quadratic polynomial format \(ax^2 + bx + c\). Depending on the values of \(a, b,\) and \(c\), it can usually be factored into two binomials of the form \((px + q)(rx + s)\).

Here's a basic approach to start factoring:

• Look for the greatest common factor (GCF) across all terms; factor it out if possible.
• For quadratic polynomials, check if they can be expressed in the form \((x + m)(x + n)\) where \(m\) and \(n\) are numbers that add to \(b\) and multiply to \(c\).
• Use special patterns, such as the difference of squares: \(a^2 - b^2 = (a + b)(a - b)\).

Once factored, these expressions can be used for simplifications in rational expressions, a vital step in managing complex algebraic fractions.

Simplification

Simplification in algebra typically involves reducing expressions to their simplest form while maintaining equivalence with the original expression. When working with rational expressions, which are fractions where both the numerator and denominator are polynomials, we aim to cancel out common factors.

Consider a common process:

• Factor both the numerator and the denominator, as seen in our example where \(x^2 - 3x - 4\) becomes \((x-4)(x+1)\).
• Identify and cancel any matching factors present in both the numerator and the denominator. In the exercise solution, the factor \(x-4\) is present in both parts and therefore gets canceled, leaving us with the simplified expression \(\frac{x-1}{x-3}\).

By following this systematic approach, complex fractions turn into more manageable expressions. But remember, only factors (not terms) can be simplified this way, ensuring we maintain an expression's original value.

Algebraic Fractions

Algebraic fractions involve expressions with polynomials in their numerators, denominators, or both. Simplifying these requires skills in both factoring and the laws of fractions. Recognizing these expressions' characteristics can make the task far less daunting.

Key points to note when working with algebraic fractions:

• Always begin by factoring. The goal is to identify factors in both the numerator and the denominator that can be canceled.
• Ensure the denominator is never zero. An expression is undefined when its denominator equals zero.
• The principles of fractions still hold. For example, when dividing fractions, you "invert and multiply." When adding or subtracting, ensure like terms (common denominators) are used.

Practicing these principles on a range of problems helps solidify understanding of rational expressions. Over time, the processes will become intuitive as you develop proficiency in factorization and simplification.