Problem 42 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{x^{2}-10 x+21}{x^{2}-6 x-7} $$

Step-by-Step Solution

Answer

Question: Simplify the given rational expression to the lowest terms: $$\frac{x^2 - 10x + 21}{x^2 - 6x - 7}$$ Answer: The simplified rational expression is: $$\frac{x-3}{x+1}$$

Step 1: Factor the numerator and the denominator

We will factor both the quadratic expressions in the numerator and the denominator as follows: Numerator: $$ x^{2}-10x+21 = (x-3)(x-7) $$ Denominator: $$ x^{2}-6x-7 = (x-7)(x+1) $$ So the given expression can be rewritten as: $$ \frac{(x-3)(x-7)}{(x-7)(x+1)} $$

Step 2: Cancel out common factors

We have a common factor of (x-7) in both the numerator and the denominator. Cancelling out the common factor, we get: $$ \frac{(x-3)\cancel{(x-7)}}{\cancel{(x-7)}(x+1)} $$

Step 3: Write the simplified expression

Now, we can write the simplified rational expression: $$ \frac{x-3}{x+1} $$ So, the given rational expression simplified to lowest terms is: $$ \frac{x-3}{x+1} $$

Key Concepts

Factoring Quadratics

When dealing with quadratics, especially in rational expressions, factoring is a key step. Quadratics are polynomials of the form \( ax^2 + bx + c \). To factor a quadratic means to express it as a product of two binomials. For example, the quadratic \( x^2 - 10x + 21 \) is factored into \((x-3)(x-7)\). This means that if you were to expand these binomials, you would get back the original quadratic equation.

Here is a simple process:

• Look for two numbers that multiply to give \( c \) (the constant term) and add to form \( b \) (the coefficient of the linear term \( x \)).
• If such pairs are found, the quadratic can be expressed as \((x-m)(x-n)\) where \( m \) and \( n \) are those numbers.

Factoring is crucial because it reveals the zeros of the polynomial and simplifies the rational expression for further operations.

Simplifying Fractions

In mathematics, simplifying fractions means reducing fractions to their lowest terms, where the numerator and the denominator share no common factors other than 1. This step is crucial in simplifying rational expressions.

After factoring the numerator and denominator, as demonstrated in our example, where the expression \( \frac{(x-3)(x-7)}{(x-7)(x+1)} \) was simplified, we're able to see the structure of the fraction more clearly. The process makes it easy to identify and cancel common factors, leading to an expression that is easier to work with.

Always remember:

• Simplified fractions make solving equations simpler.
• It eases comparison between the fractions.
• Understanding becomes straightforward as unnecessary parts are removed.

Simplification is not just a matter of cosmetic preference; it's a fundamental mathematical practice that clarifies calculations.

Canceling Common Factors

Once a fraction is fully factored, canceling common factors is the next logical step. This is where the numerator and denominator have a common factor, and by dividing both by this term, the fraction is reduced to a simpler equivalent.

For example, after factoring, our expression \( \frac{(x-3)(x-7)}{(x-7)(x+1)} \) had \((x-7)\) as a common factor. This factor can be 'canceled' in both the numerator and the denominator to simplify the expression to \( \frac{x-3}{x+1} \).

Key points to remember:

• Only cancel factors, not terms. This means entire binomials or monomials, not just parts of them.
• Each factor needs to be present in both the numerator and the denominator to cancel it effectively.
• Cancellation helps in ensuring the rational expression is in its simplest form.

By mastering this technique, students can tackle a variety of algebraic problems with greater ease and accuracy.