Problem 104 - Algebra and Trigonometry

Question

\(g(x)=\frac{x-6}{x^{2}-36}\) (Section 3.5, Example 1)

Step-by-Step Solution

Answer

The simplified form of the function is \(g(x)=\frac{1}{x+6}\).

Step 1: Identifying the Difference of Squares

Notice that the denominator \(x^{2}-36\) can be expressed as \(a^{2}-b^{2}\), which, analyzing it, represents a difference of squares. Where \(a^{2}\) is \(x^{2}\) and \(b^{2}\) is \(36\)

Step 2: Factorising the Denominator

We factorize \(x^{2}-36\) according to the rule \(a^{2}-b^{2} = (a-b)(a+b)\) given that it is a difference of squares. Thus, \(x^{2}-36\) can be factored into \((x-6)(x+6)\)

Step 3: Simplifying the function

Now, the function becomes \(g(x)=\frac{x-6}{(x-6)(x+6)}\). We notice that \(x-6\) can be cancelled out from the numerator and the denominator. Hence, the function simplifies into \(g(x)=\frac{1}{x+6}\)

Key Concepts

Difference of Squares

When working with rational expressions, one useful concept to understand is the difference of squares. This is specifically helpful because it streamlines the factoring process. An expression written as \(a^2 - b^2\) is what we call a difference of squares. It means we have a subtraction between two perfect square terms.

You can identify a difference of squares because it will always have two terms, connected by a subtraction sign. Both terms are perfect squares, meaning they are the square of some number or variable. For example, \(x^2 - 36\) fits the pattern, where \(x^2\) is the square of \(x\) and \(36\) is the square of 6. Knowing this, you can then apply the formula:

• If you have \(a^2 - b^2\), it factors into \((a-b)(a+b)\).

This knowledge transforms otherwise complex expressions into very manageable equations, simplifying your work in proceeding steps.

Factoring Polynomials

Once you've identified the difference of squares, as in \(x^2 - 36\), the next step is to factor the polynomial. Factoring polynomials is like breaking them into parts that are easier to work with. For a difference of squares, this is straightforward because you can immediately apply the formula:

• For \(x^2 - 36\), use \(a = x\) and \(b = 6\), then \(x^2 - 36 = (x-6)(x+6)\).

This process involves just recognizing each squared term and rewriting the expression as a product of two binomials. Factoring polynomials this way is crucial because it lays the ground for simplifying the rational expressions. After factoring, the expression is already much easier to handle.

Canceling Common Factors

After the polynomial is factored, the next step in simplifying rational expressions is to cancel out any common factors in the numerator and denominator. This is like reducing fractions in regular arithmetic.

Looking at the example, \(rac{x-6}{(x-6)(x+6)}\), we see \(x-6\) is both in the numerator and as part of the denominator when rewritten as \(x^2-36 = (x-6)(x+6)\).

• Because \(x-6\) appears in both places, we can "cancel" it out, simplifying our rational expression to \(rac{1}{x+6}\).

Always remember to only cancel out identical terms, ensuring the final expression remains equivalent to the original. This step is key for achieving the simplest form of the expression you’re working with.