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$$ \frac{5 r^{2}+7 r-33}{r^{2}-49}-\frac{4 r^{2}+5 r+30}{r^{2}-49} $$

   $$
\frac{5 r^{2}+7 r-33}{r^{2}-49}-\frac{4 r^{2}+5 r+30}{r^{2}-49}
$$

Intermediate Algebra

Intermediate Algebra

Lynn Marecek 2nd Edition

Chapter 7, Problem 89

Instant Answer

Solved by Expert Victor Salazar

09/24/2022

Step 1

Since they have the same denominator, we can simply subtract the numerators: $$ \frac{5 r^{2}+7 r-33}{r^{2}-49}-\frac{4 r^{2}+5 r+30}{r^{2}-49} = \frac{(5 r^{2}+7 r-33) - (4 r^{2}+5 r+30)}{r^{2}-49} $$ Show more…

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$$ \frac{5 r^{2}+7 r-33}{r^{2}-49}-\frac{4 r^{2}+5 r+30}{r^{2}-49} $$

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Key Concepts

Domain Restrictions

When working with rational expressions, it is important to consider domain restrictions, which are the values that make the denominator zero. Identifying these values is necessary to ensure that any simplification or transformation of the expression remains valid within its domain.

Factorization of Polynomials

Factorization is the process of writing a polynomial as a product of its factors. It is particularly useful in simplifying rational expressions by revealing common factors in the numerator and denominator that may cancel out, thereby reducing the expression to its simplest form.

Polynomial Operations

Subtracting rational expressions often involves performing polynomial subtraction on the numerators. This process includes distributing negative signs, combining like terms, and simplifying the resulting polynomial, which is a fundamental skill in algebra.

Common Denominator

When subtracting or adding rational expressions, having a common denominator is crucial because it allows the direct subtraction of the numerators. This step simplifies the process by focusing on combining like terms in the numerators while retaining the shared denominator.

Rational Expressions

A rational expression is a fraction in which both the numerator and the denominator are polynomials. Understanding how to manipulate these expressions, including addition, subtraction, multiplication, and division, is essential for solving algebraic equations and simplifying formulas.

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