Simplify. (x^5-x^3+x^2-1-(x^3-1)(x+1)^2)/((x^2-1)^2) |...

Name: Problem 101, Chapter 6: Intermediate Algebra : Concepts and Applications
Uploaded: 2020-07-09T16:07:21-08:00
Duration: 2 min 42 s
Channel: Ankit Gupta
Description: Simplify. (x^5-x^3+x^2-1-(x^3-1)(x+1)^2)/((x^2-1)^2)

Key Concepts

Algebraic Expansion

This concept involves multiplying polynomials using the distributive property to expand products into sums of terms. For example, when you expand expressions like (x^3 - 1)(x + 1)^2, you multiply each term in one polynomial by each term in the other, resulting in a polynomial that can then be simplified.

Combining Like Terms

After expanding the polynomials, the next step is to combine like terms – terms with the same degree – to simplify the expression. This is essential for reducing the polynomial into its simplest form before further manipulation.

Polynomial Factorization

Factoring polynomials is the process of writing a polynomial as the product of its factors. Techniques such as factoring by grouping or recognizing common patterns are used to identify and extract factors, which often simplifies the expression and reveals cancellations.

Difference of Squares

This technique is a specific method of factorization used when an expression is of the form a^2 - b^2, which factors into (a + b)(a - b). Recognizing that x^2 - 1 fits this pattern is useful because it allows one to factor the denominator into simpler binomials.

Cancellation of Common Factors

Once both the numerator and denominator are factored, any common factors between them can be cancelled as long as they are not zero. This reduction is a key step in simplifying rational expressions, leading to a simpler and more interpretable result.

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