Divide and simplify. See Example 7 (x^2-3 x)/(x^3-27) ÷(2 x)/(2 x^2+6 x+18)

Name: Problem 54, Chapter 6: Intermediate Algebra
Uploaded: 2020-07-16T20:12:02-08:00
Duration: 56 s
Channel: Amy Jiang
Description: Divide and simplify. See Example 7 (x^2-3 x)/(x^3-27) ÷(2 x)/(2 x^2+6 x+18)

step 1 Okay, so we want to simplify the following. So let's start by factoring each term. So this term,

Divide and simplify. See Example 7 (x^2-3 x)/(x^3-27) ÷(2...

Key Concepts

Domain Restrictions

When simplifying rational expressions, it is crucial to consider the values that would make any denominator zero. These restrictions, which may not be obvious after simplification, define the domain of the original expression and ensure mathematical validity throughout the problem-solving process.

Canceling Common Factors

After factoring both the numerator and denominator in rational expressions, any factors that are present in both can be canceled out. This simplifies the expression considerably but requires careful attention to ensure that cancellation is valid and that none of the canceled factors are equal to zero.

Polynomial Factoring

Factoring polynomials is the process of breaking down a polynomial into a product of simpler polynomials or binomials. This includes techniques such as factoring out the greatest common factor, factoring quadratics, and applying formulas like the difference of squares or cubes. Mastery of these techniques is essential for simplifying rational expressions.

Division of Rational Expressions

Dividing one rational expression by another involves multiplying the first expression by the reciprocal of the second. This process is key because it transforms a division problem into a multiplication one, making it easier to simplify the resulting expression after performing appropriate factorizations.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are both polynomials. They form an important part of algebra and require special techniques when performing operations such as addition, subtraction, multiplication, and division, all while keeping track of conditions that prevent division by zero.

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