Problem 54 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{x^{2}-16}{x^{2}-8 x+16} $$

Step-by-Step Solution

Answer

The expression simplifies to \(x+4\), where \(x \neq 4\).

Step 1: Factor the Numerator

The numerator of the expression is \(x^2 - 16\), which is a difference of squares. Recall that \(a^2 - b^2 = (a-b)(a+b)\). Thus, \(x^2 - 16 = (x-4)(x+4)\).

Step 2: Factor the Denominator

The denominator is \(x^2 - 8x + 16\). This is a perfect square trinomial. Recall that \(a^2 - 2ab + b^2 = (a-b)^2\). Here, \(x^2 - 8x + 16 = (x-4)^2\).

Step 3: Write the Expression with Factored Parts

After factoring both the numerator and denominator, we rewrite the expression as \(\frac{(x-4)(x+4)}{(x-4)^2}\).

Step 4: Simplify the Fractions

In the fraction \(\frac{(x-4)(x+4)}{(x-4)^2}\), the \((x-4)\) terms in the numerator and denominator can be cancelled out, as long as \(x eq 4\) to avoid division by zero. This simplifies to \(\frac{x+4}{x-4}\).

Step 5: Final Simplified Expression

After canceling the common term in the numerator and denominator, the expression simplifies to \(x + 4\). Always remember the restriction \(x eq 4\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a powerful technique used to simplify expressions. It involves breaking down a polynomial into simpler "factor" terms that, when multiplied together, give you the original polynomial. This can make equations easier to work with or solve.
For example, in our exercise, the numerator \(x^2 - 16\) is a difference of squares, and it can be rewritten as \((x - 4)(x + 4)\).
The key is to recognize patterns:

• A difference of squares looks like \(a^2 - b^2\), which factors into \((a-b)(a+b)\).
• Recognizing such patterns quickly can help you factor polynomials efficiently.

By mastering factoring, you prepare yourself for more complex algebra problems. It's all about spotting patterns and simplifying.

Simplification of Fractions

Simplifying fractions means reducing them to their simplest form. This involves canceling common factors from the numerator and the denominator.
In our case, after factoring the numerator and denominator, the expression becomes \(\frac{(x-4)(x+4)}{(x-4)^2}\). Here, the common factor \((x-4)\) appears in both the top and bottom.
By removing or cancelling this common term, we simplify the expression to \(\frac{x+4}{x-4}\).
A crucial reminder: ensure that any values which make the denominator zero are excluded, such as \(x eq 4\), since division by zero is undefined.
Simplification not only makes expressions easier to understand, but also helps solve equations faster.

Difference of Squares

The difference of squares is a special type of expression that takes the form \(a^2 - b^2\). It’s quite unique because it can always be factored into \((a-b)(a+b)\).
This pattern allows for quick and straightforward simplification of many polynomial expressions.
In the provided exercise, \(x^2 - 16\) is a perfect example. Here, 16 is the square of 4, and so \(x^2 - 16\) becomes \((x-4)(x+4)\).
This technique is especially handy when dealing with polynomial equations and simplifying complex algebraic expressions, ensuring you maintain balance by applying it consistently wherever applicable.

Perfect Square Trinomials

Perfect square trinomials are algebraic expressions in the form \(a^2 - 2ab + b^2\), which factor down neatly into \((a-b)^2\).
They represent squares of binomials and allow for ease of computation especially when they appear in the denominator, as seen in our exercise with \(x^2 - 8x + 16\).
This expression is a perfect square trinomial because it can be rewritten as \((x-4)^2\).
Recognizing these trinomials can significantly simplify your work with polynomials.
Knowing how to factor them quickly ensures that algebra problems are less daunting, and creates more efficient avenues for solving problems effortlessly.