Perform the indicated operations and simplify. (x^3-8)/(x^2+2 x-8) ÷(x^2+2 x+4)/(x^2+7 x+12)

Name: Problem 51, Chapter 7: Intermediate Algebra: Connecting Concepts through Applications
Uploaded: 2022-03-17T19:43:44-08:00
Duration: 2 min 1 s
Channel: Alison Rodriguez
Description: Perform the indicated operations and simplify. (x^3-8)/(x^2+2 x-8) ÷(x^2+2 x+4)/(x^2+7 x+12)

step 1 This problem here, we're going to be doing division. So when you divide, we're going to need to

Perform the indicated operations and simplify....

Key Concepts

Cancellation in Rational Expressions

After factoring both the numerator and denominator, any common factors can be canceled, provided they are not equal to zero. This step is crucial for reducing expressions to their simplest form while being mindful of restrictions in the domain of the variable.

Quadratic Factoring

This technique is used to factor second-degree polynomials, typically by finding two numbers that multiply to give the constant term and add to give the coefficient of the linear term. Mastery of quadratic factoring is essential for breaking down rational expressions and simplifying them by canceling common factors.

Division of Rational Expressions

This concept involves dividing one fraction by another, which is accomplished by multiplying by the reciprocal of the divisor. Understanding this process is important for correctly simplifying complex rational expressions involving polynomials.

Polynomial Factorization

This concept involves expressing a polynomial as the product of simpler polynomials that, when multiplied together, give the original polynomial. Factoring is fundamental for simplifying expressions, solving equations, and performing operations such as division of rational expressions.

Difference of Cubes

The difference of cubes is a specific factorization formula applied to expressions in the form a³ - b³, which can be factored as (a - b)(a² + ab + b²). This formula simplifies complex cubic expressions and is a key tool when handling algebraic fractions.

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