Multiply and, if possible, simplify. (t^3-27)/(t^4-9 t^2) ·(t^5-6 t^4+9 t^3)/(t^2+3 t+9)

Name: Problem 54, Chapter 6: Intermediate Algebra : Concepts and Applications
Uploaded: 2020-07-13T03:37:57-08:00
Duration: 3 min 42 s
Channel: Ankit Gupta
Description: Multiply and, if possible, simplify. (t^3-27)/(t^4-9 t^2) ·(t^5-6 t^4+9 t^3)/(t^2+3 t+9)

step 1 In this question we have to apply a formula a cube minus p cube for t q minus 27 t cube minus 27

Multiply and, if possible, simplify. (t^3-27)/(t^4-9 t^2)...

Key Concepts

Domain Considerations

When working with rational expressions, it's important to identify the domain by determining the values that make any denominator zero. These restrictions help to ensure that any operations or simplifications performed are valid within the appropriate set of values.

Simplification and Cancellation

After multiplying rational expressions, simplifying the resulting expression is usually done by factoring both the numerator and denominator and canceling out any common factors. This step is crucial for obtaining the simplest form of an expression while ensuring the mathematical integrity of the equation.

Difference of Cubes Factorization

Difference of cubes is a specialized factoring technique used when a term can be written as a cube minus another cube. The formula a³ - b³ = (a - b)(a² + ab + b²) is vital for recognizing and simplifying such expressions in algebra.

Polynomial Factoring

In algebra, factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This process is essential for simplifying expressions and solving equations by breaking complex expressions into factors that can be more easily manipulated or canceled.

Multiplication of Rational Expressions

Multiplying rational expressions requires multiplying the numerators together and the denominators together. This process is central in algebra and leads to a new rational expression that can often be further simplified by factoring and canceling common factors.

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