Problem 46 - Algebra A Combined Function

Question

Rewrite each rational expression as an equivalent rational expression with the given denominator. $$ \frac{6 m-5}{3 x^{2}-9}=\frac{\underline{\phantom{xx}}}{12 x^{2}-36} $$

Step-by-Step Solution

Answer

The equivalent expression is \(\frac{24m - 20}{12x^2 - 36}\).

Step 1: Identify the Relationship Between the Denominators

First, examine the denominators of the given rational expressions. The original expression has a denominator of \(3x^2 - 9\), and the desired denominator is \(12x^2 - 36\). Notice that \(12x^2 - 36\) is 4 times \(3x^2 - 9\). This suggests that the original denominator can be multiplied by 4 to obtain the new denominator.

Step 2: Express the Original Denominator in Terms of the New Denominator

To verify, factor the original denominator \(3x^2 - 9\). It can be factored as \(3(x^2 - 3)\). Now factor the new denominator: \(12x^2 - 36 = 12(x^2 - 3)\). Rewrite it as \(4 \times 3(x^2 - 3)\). This confirms that \(12x^2 - 36\) is \(4\) times \(3x^2 - 9\).

Step 3: Scale the Numerator

Since the denominator is multiplied by 4, we need to multiply the numerator \(6m - 5\) by 4 to maintain equivalence. Therefore, the new numerator is \(4 \times (6m - 5) = 24m - 20\).

Step 4: Write the Equivalent Expression

Now that both the numerator and the denominator are scaled appropriately, the equivalent rational expression is given by: \(\frac{24m - 20}{12x^2 - 36}\).

Key Concepts

Rational Expressions

A rational expression is like a fraction, but instead of numbers, we use polynomials. A polynomial could be something as simple as just a constant (like 5) or something more complex, like expression with variables (such as \( 3x^2 - 9 \)).

• Rational expressions have a numerator (the top part) and a denominator (the bottom part), both of which are polynomials.
• Just like fractions, rational expressions need to be simplified when possible. This often involves factoring.

To rewrite a rational expression with a different denominator, think about what you need to multiply the old denominator by to get the new one. You apply that same factor to the numerator as well. This process keeps the value of the rational expression the same, just as it does when adjusting fractions.

Factoring

Factoring is a crucial math skill, especially for dealing with rational expressions. It's the process of breaking down a complex expression into simpler parts (factors) that multiply to make the original expression.

• For example, the polynomial \( 3x^2 - 9 \) can be factored as \( 3(x^2 - 3) \) by pulling out the common factor, which is 3.
• Similarly, \( 12x^2 - 36 \) is factored into \( 12(x^2 - 3) \), which can be broken down further into \( 4 \times 3(x^2 - 3) \).

Factoring helps us see how two expressions are connected, especially when trying to rewrite expressions with a different denominator, as all related operations are based on these factored forms.

Equivalent Expressions

Equivalent expressions are different representations of the same mathematical idea. They're "equal" because they give the same result for every value of the variable.

• To create an equivalent expression, it's important to apply the same mathematical operation to both the numerator and the denominator of a rational expression.
• In the given exercise, multiplying both parts of the original expression by 4 creates an equivalent expression: \(\frac{24m - 20}{12x^2 - 36}\), from \(\frac{6m - 5}{3x^2 - 9}\).

Understanding and finding equivalent expressions is vital for solving algebraic problems, as it allows you to work with an expression in a desired form, often simplifying complex operations.