Problem 84 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{x^{2}-3 x+2}{x^{2}-4 x+3} \div(x-3) $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{x^{2}-3 x+2}{x^{2}-4 x+3} \div (x-3)\). Answer: \(x-2)

Step 1: Turn the division into multiplication

To perform the division operation, we need to convert it into multiplication by taking the reciprocal of the second term. $ \frac{x^{2}-3 x+2}{x^{2}-4 x+3} \div (x-3) $ becomes $ \frac{x^{2}-3x+2}{x^{2}-4x+3} \cdot \frac{1}{x-3} $

Step 2: Factorize the expressions if possible

We can factorize the expressions in the numerator and denominator to simplify the fractions further. $$ (x^2-3x+2) = (x-2)(x-1) $$ $$ (x^2-4x+3) = (x-3)(x-1) $$ So, our expression becomes $ \frac{(x-2)(x-1)}{(x-3)(x-1)} \cdot \frac{1}{x-3} $

Step 3: Cancel common factors

Now, we can cancel the common factors in the expression. $ = \frac{(x-2)\cancel{(x-1)}}{\cancel{(x-3)}\cancel{(x-1)}} \cdot \frac{1}{\cancel{x-3}} \quad (\text{since } x-3, x-1 \neq 0) $ Now, we have: $ = \frac{x-2}{1} $

Step 4: Simplify the expression

Since there is no need for a denominator, our simplified expression is just the numerator. \(x-2\) So, the given expression simplifies to \(x-2\).

Key Concepts

Algebraic Fractions

Algebraic fractions are just like regular fractions, but instead of numbers, they have algebraic expressions in the numerator and the denominator. One key skill in working with algebraic fractions is the ability to convert division problems into multiplication problems. This usually involves flipping the second fraction (taking the reciprocal) and changing the division sign to a multiplication sign.

For example, the division of two algebraic expressions, \( \frac{x^2-3x+2}{x^2-4x+3} \div (x-3) \), is converted to multiplication by taking the reciprocal of \(x-3\) to get \( \frac{x^2-3x+2}{x^2-4x+3} \cdot \frac{1}{x-3} \). This step transforms the division of fractions into a more familiar operation—multiplication—which simplifies the process of reducing the expression.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler 'factor' expressions that, when multiplied together, give back the original polynomial. It's a bit like finding what ingredients went into a cake so that you can understand it better and, if necessary, modify the recipe.

For example, the quadratic polynomial \(x^2-3x+2\) can be factored into \(x-2\) and \(x-1\). Likewise, \(x^2-4x+3\) factors into \(x-3\) and \(x-1\). Recognizing these factors is crucial because it allows us to cancel out common terms in algebraic fractions, which greatly simplifies the expression. In our initial expression, this factoring led to the cancellation of \(x-1\) and \(x-3\), leaving us with a simpler form \(\frac{x-2}{1}\).

Simplifying Expressions

The goal of simplifying expressions is to rewrite them in the simplest form possible, where 'simplest form' means no common factors, and the expression can't be made any 'shorter' or more compact. Simplification can involve canceling common terms, combining like terms, or carrying out operations.

After factoring and canceling out common factors, in our example, we have \(\frac{x-2}{1}\). Since any number divided by 1 is itself, we simplify our expression to just \(x-2\). Always look for opportunities to simplify expressions; it's not just about making the math look nicer but also about making your calculations easier and error-free.