Divide and, if possible, simplify. (z^2-2 z+1)/(z^2-1) ÷(4 z^2-z-3)

Name: Problem 70, Chapter 7: Elementary and Intermediate Algebra
Uploaded: 2020-05-28T21:44:06-08:00
Duration: 1 min 43 s
Channel: Jeffrey Morris
Description: Divide and, if possible, simplify. (z^2-2 z+1)/(z^2-1) ÷(4 z^2-z-3)

step 1 Okay, so I went ahead and factored out all of the polynomials on the left down the right stat of

Divide and, if possible, simplify. (z^2-2 z+1)/(z^2-1) ÷(4...

Key Concepts

Polynomial Factorization

This involves rewriting a polynomial as a product of simpler polynomials or factors. It is a fundamental technique used in algebra to simplify expressions, solve equations, and reveal structures within polynomials. Factoring helps in identifying common terms and potential cancellations in rational expressions.

Division of Rational Expressions

Dividing one rational expression by another is accomplished by multiplying the first expression by the reciprocal of the second. This transformation converts division into multiplication, which allows for techniques like factoring and cancellation to be applied more readily in the simplification process.

Cancellation of Common Factors

When simplifying complex rational expressions, it is important to identify and cancel common factors in the numerator and denominator. This step often leads to a simpler form, but one must be cautious to ensure that the values for which the original denominators are zero are excluded from the domain.

Domain Restrictions

In any rational expression, the values that make the denominator zero must be excluded from the set of permissible values. Establishing these restrictions is crucial since cancellation of factors does not remove the original constraints imposed by the denominators. This ensures the simplified expression remains equivalent to the original expression under its defined domain.

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