Question

Determine the polynomial to be placed in the shaded area to give a true statement. $$ \frac{}{x^{2}+2 x-15}=\frac{1}{x-3} $$

Name: Problem 79, Chapter 6: Intermediate Algebra for College Students
Uploaded: 2022-02-27T04:29:29-08:00
Duration: 32 s
Channel: Erika Bustos
Description: Determine the polynomial to be placed in the shaded area to give a true statement. ()/(x^2+2 x-15)=(1)/(x-3)

   Determine the polynomial to be placed in the shaded area to give a true statement.
$$
\frac{}{x^{2}+2 x-15}=\frac{1}{x-3}
$$

Intermediate Algebra for College Students

Allen R. Angel;… 10th Edition

Chapter 6, Problem 79

Instant Answer

Solved by Expert Erika Bustos

02/27/2022

Step 1

We know that one factor is (x-3), so let's find the other factor. (x-3)(x+a) = x^2 + 2x - 15 Expanding this, we get: x^2 + ax - 3x - 3a = x^2 + 2x - 15 Now, we can compare the coefficients of the powers of x: a - 3 = 2 (coefficient of x) -3a = -15 (constant Show more…

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Determine the polynomial to be placed in the shaded area to give a true statement. $$ \frac{}{x^{2}+2 x-15}=\frac{1}{x-3} $$

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Key Concepts

Factoring Quadratic Polynomials

This concept involves rewriting a quadratic expression as a product of two binomials, which simplifies the manipulation of expressions and equations. Recognizing factorable forms is crucial for handling rational expressions and for canceling common factors in an equation.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to work with these expressions—including performing operations, simplifying them through factoring, and finding equivalent expressions—is a key component in algebra.

Clearing Denominators

Clearing denominators is a technique used to eliminate fractions in an equation by multiplying both sides by the common denominator. This process transforms a rational equation into a polynomial equation, making it simpler to identify and solve for unknown quantities.

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