Question

Simplify each expression. $$ \frac{5}{x^{2}-3 x-28}+\frac{7}{2 x-14} $$

   Simplify each expression.
$$
\frac{5}{x^{2}-3 x-28}+\frac{7}{2 x-14}
$$

Algebra 2

Holliday, Luchin,… 1st Edition

Chapter 8, Problem 30

Instant Answer

Solved by Expert Ashley High

03/18/2020

Step 1

The first denominator $x^{2}-3 x-28$ can be factored into $(x-7)(x+4)$ and the second denominator $2x-14$ can be factored into $2(x-7)$. So, the expression becomes: $$ \frac{5}{(x-7)(x+4)}+\frac{7}{2(x-7)} $$ Show more…

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Simplify each expression. $$ \frac{5}{x^{2}-3 x-28}+\frac{7}{2 x-14} $$

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Key Concepts

Simplification by Canceling Common Factors

Once rational expressions are combined, any common factors in the numerator and denominator can be canceled, provided they are not equal to zero. This step is crucial in reducing the expression to its simplest form and must be done while keeping track of domain restrictions to ensure the mathematical validity of the simplification.

Factoring Polynomial Expressions

This concept involves rewriting a polynomial as a product of its factors. In the context of simplifying rational expressions, factoring allows one to break down complex expressions into simpler linear or quadratic terms, which can then be more easily manipulated or canceled with similar factors in the numerator or denominator.

Rational Expressions

Rational expressions are fractions that have polynomials in the numerator and the denominator. Simplifying these expressions often requires techniques such as factoring and finding a common denominator, as well as careful consideration of restrictions on the variable that arise from values that make the denominator zero.

Finding a Common Denominator

To add or subtract rational expressions, it is necessary to rewrite them so they have the same denominator. This typically involves factoring the denominators first and then determining the least common denominator that accommodates all factors from each rational expression.

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