Problem 32 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{x^{2}-14 x+49}{6 x^{2}-37 x-35}\)

Step-by-Step Solution

Answer

\( \frac{x - 7}{6x + 5} \)

Step 1: Identify the Numerator and Denominator

The given rational expression is \( \frac{x^2 - 14x + 49}{6x^2 - 37x - 35} \). Here, the numerator is \( x^2 - 14x + 49 \) and the denominator is \( 6x^2 - 37x - 35 \).

Step 2: Factor the Numerator

The numerator \( x^2 - 14x + 49 \) is a perfect square trinomial. It can be factored as \((x - 7)(x - 7) \) or simply \((x - 7)^2 \).

Step 3: Factor the Denominator

The denominator \( 6x^2 - 37x - 35 \) needs to be factored. First, look for two numbers whose product is \( 6 \times -35 = -210 \) and whose sum is \(-37\). Those numbers are \(-42\) and \(5\). Rewrite the middle term and factor by grouping:\( 6x^2 - 42x + 5x - 35 = 6x(x - 7) + 5(x - 7) = (6x + 5)(x - 7) \).

Step 4: Simplify the Rational Expression

The fully factored expression is \( \frac{(x - 7)(x - 7)}{(6x + 5)(x - 7)} \). Cancel the common term \((x - 7)\) from the numerator and the denominator, resulting in \( \frac{x - 7}{6x + 5} \).

Key Concepts

Factoring Trinomials

Trinomials are expressions composed of three terms. Factoring these expressions is a key step in simplifying many algebraic functions, especially rational expressions. When we talk about factoring trinomials, we are finding two binomials whose product equals the given trinomial. For example, let's look at our expression's numerator:

• The trinomial is \(x^2 - 14x + 49\).

In this particular case, it represents a perfect square trinomial (which I'll explain in more detail in the next section), allowing us to write it as \((x - 7)(x - 7)\). The process usually involves:

• Determining if the trinomial can be factored at all, by either inspection, using the quadratic equation, or identifying unique patterns.
• Once the trinomial is factored, it often simplifies equations significantly when dealing with rational expressions.

The key to mastering this skill is practice and recognizing the various forms and patterns trinomials can take.

Perfect Square Trinomials

A perfect square trinomial is a special type of trinomial that results from squaring a binomial. In mathematical terms, it means the expression can be written as \((a - b)^2\) or \((a + b)^2\). Such trinomials often look like \(a^2 \pm 2ab + b^2\).

• In our case, \(x^2 - 14x + 49\) is a perfect square since it becomes \((x - 7)^2\).

Identifying a perfect square trinomial is useful because it simplifies both factoring and subsequent algebraic manipulation. Here’s a quick way to recognize one:

• The first and last terms are perfect squares.
• The middle term is twice the product of the square roots of the first and last terms.

Recognizing these can save time during calculations and allow simplifications with more confidence.

Factoring by Grouping

Factoring by grouping is a technique used to factor complex polynomials. When a standard approach to factoring isn’t evident, grouping pairs of terms can reveal a common factor. Let’s take an example from the denominator in our given expression:

• \(6x^2 - 37x - 35\) initially doesn’t yield an easy factorization.

By rewriting it using two numbers that multiply to \(-210\) (\((-42) \times 5 = -210\)) and add to \(-37\):

• Rewritten, it is \(6x^2 - 42x + 5x - 35\).
• Group it into two pairs: \((6x^2 - 42x) + (5x - 35)\).

For each pair, extract the common factor. Let's see the steps:

• For \(6x^2 - 42x\), factor out \(6x\) to get \(6x(x - 7)\).
• For \(5x - 35\), factor out \(5\) to get \(5(x - 7)\).
• We now have \((6x + 5)(x - 7)\).

Factoring by grouping can simplify complex expressions considerably, and recognizing when to use this method develops through practice and experimentation.

Rational Expression Simplification

Rational expression simplification is all about finding simpler forms of expressions where possible. In the context of algebra, a rational expression is like a fraction where both the numerator and the denominator are polynomials. Simplifying these expressions often involves factoring. Let’s see the steps used in the given problem:

• Once the numerator and denominator are factored, we have \(\frac{(x-7)^2}{(6x+5)(x-7)}\).
• Notice the common term \((x-7)\) in both numerator and denominator. Since it’s not affecting the essential value of the expression aside from when \(x=7\), it can be canceled.

After canceling, the expression reduces to \(\frac{x-7}{6x+5}\). Keep in mind that simplification doesn’t change the expression's inherent "value." It may, however, introduce restrictions such as setting conditions where the original might be undefined (e.g., \(x eq 7\) in this case).