In Exercises 15–32, multiply or divide as indicated. (x2−4)/(x2+3x−10) ÷ (x2+5x+6)/(x2+8x+15)

Ch. P - Fundamental Concepts of Algebra

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

All textbooks

Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problem 30

Chapter 1, Problem 30

In Exercises 15–32, multiply or divide as indicated. (x²−4)/(x²+3x−10) ÷ (x²+5x+6)/(x²+8x+15)

Verified step by step guidance

Rewrite the division of fractions as multiplication by the reciprocal. This means flipping the second fraction. The expression becomes: ((x^2−4)/(x^2+3x−10)) * ((x^2+8x+15)/(x^2+5x+6)).

Factorize all the polynomials in the numerators and denominators. For example: x^2−4 is a difference of squares and factors as (x−2)(x+2). Similarly, factorize x^2+3x−10, x^2+8x+15, and x^2+5x+6.

After factoring, the expression will look like: (((x−2)(x+2))/((x+5)(x−2))) * (((x+3)(x+5))/((x+3)(x+2))).

Cancel out any common factors in the numerators and denominators across the multiplication. For instance, (x−2), (x+2), (x+5), and (x+3) may cancel out depending on their positions.

Write the simplified expression after canceling the common factors. This will be the final simplified result of the given problem.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including simplifying, multiplying, and dividing them, is crucial for solving problems involving them. For example, recognizing that (x^2−4) can be factored into (x−2)(x+2) helps in simplifying the expression.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential for simplifying rational expressions, as it allows for cancellation of common factors. For instance, the polynomial x^2 + 3x - 10 can be factored into (x + 5)(x - 2), which simplifies the division process.

Division of Rational Expressions

Dividing rational expressions involves multiplying by the reciprocal of the divisor. This means that to divide (x^2−4)/(x^2+3x−10) by (x^2+5x+6)/(x^2+8x+15), you first factor both expressions and then multiply by the reciprocal of the second expression. This process is fundamental in solving complex rational equations.

Recommended video:

Guided course

02:58

Rationalizing Denominators

Related Practice

Textbook Question

In Exercises 15–58, find each product. (7x³+5)(x²−2)

views

Textbook Question

Simplify each exponential expression in Exercises 23–64. $x^{- 5} \cdot x^{10}$

Textbook Question

Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √24x^4/√3x