Identify the least common denominator of each group of...

Identify the least common denominator of each group of rational expression, and rewrite each as an equivalent rational expression with the LCD as its denominator.

$\frac{7}{2 x-3}, \frac{4 x}{9-4 x^{2}}$

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Understanding these expressions involves operations such as simplification, addition, subtraction, multiplication, and division, all of which require a solid grasp of polynomial arithmetic and factoring techniques.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into a product of simpler polynomials or expressions that, when multiplied together, yield the original polynomial. This step is crucial for identifying common factors in denominators and numerators, which simplifies the process of finding a common denominator or reducing the expression.

Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest expression that can serve as a common denominator for a set of rational expressions. It is found by factoring each denominator into its irreducible factors and then taking each factor to its highest power present in any of the denominators. This concept is essential when performing addition or subtraction of rational expressions.

Rewriting Rational Expressions with a Common Denominator

Once the LCD has been identified, rewriting each rational expression with that common denominator involves multiplying the numerator and denominator of each fraction by the appropriate factors. This ensures that the fractions have the same denominator, thereby allowing for direct addition or subtraction and further manipulation of the rational expressions.

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