Problem 89 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{x^{2}-6 x+9}{81-x^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{(x-3)^2}{(9-x^2)(9+x^2)} \).

Step 1: Factor the Numerator

The numerator is given as \( x^2 - 6x + 9 \). We recognize this as a perfect square trinomial. This can be factored as \((x - 3)^2\) because \((x - 3)(x - 3) = x^2 - 6x + 9\).

Step 2: Factor the Denominator

The denominator is given as \(81 - x^4\). This can be rewritten as \((9^2 - (x^2)^2)\), which is a difference of squares. The difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\) gives us \((9 - x^2)(9 + x^2)\).

Step 3: Identify Common Factors

The factored expression is \(\frac{(x-3)^2}{(9-x^2)(9+x^2)}\). There are no common factors between the numerator and the denominator, so no further simplification is possible.

Step 4: Simplify the Expression

Since there are no common factors, the expression remains: \( \frac{(x-3)^2}{(9-x^2)(9+x^2)} \).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic expression. It arises when a binomial is multiplied by itself, meaning it's squared. This specific structure can be identified when an expression follows the pattern of

• \( a^2 - 2ab + b^2 = (a-b)^2 \)
• or \( a^2 + 2ab + b^2 = (a+b)^2 \).

In the original exercise, the numerator \( x^2 - 6x + 9 \) is a perfect square trinomial. Notice how it fits the form of \( (a-b)^2 \) with \( a = x \) and \( b = 3 \). This makes it easily factorable into \( (x-3)^2 \). Recognizing this pattern is key to simplifying such expressions. The ability to spot and factor perfect square trinomials is a valuable skill in algebra, as they frequently appear in more complex problems and equations.

Difference of Squares

The concept of the difference of squares is another fundamental algebraic idea. It involves expressions that can be rewritten as the difference between two squared terms. The standard form is \( a^2 - b^2 \), which can be factored using the formula:

• \( a^2 - b^2 = (a-b)(a+b) \).

In the exercise, the denominator \( 81 - x^4 \) aligns perfectly with this concept. Here, \( 81 \) is the square of \( 9 \) and \( x^4 \) is the square of \( x^2 \). Therefore, this expression can be rewritten as \( (9^2 - (x^2)^2) \) and factored into \( (9-x^2)(9+x^2) \). Spotting a difference of squares allows for quick factoring and simplification, and is especially useful for reducing complex expressions.

Factoring Quadratics

Factoring quadratics involves rewriting quadratic expressions as the product of two binomials. Recognizing special patterns like perfect square trinomials and the difference of squares is critical. However, not all quadratics fit these special cases, and some may require specific techniques like:

• Trial and error
• using the quadratic formula
• or completing the square.

In the exercise, both the numerator and the denominator were able to be factored using these special forms, which is not always the case in more general expressions. Once an expression is factored, identifying and canceling common factors between the numerator and denominator is crucial for simplification. However, as seen here, when there are no common factors, as in \( \frac{(x-3)^2}{(9-x^2)(9+x^2)} \), the expression is already in its simplest form. Understanding how to factor and simplify involves practice and familiarity with different methods.