Question

Simplify each expression. $\frac{b^{2}-4 b+3}{3-2 b-b^{2}}$

   Simplify each expression.
$\frac{b^{2}-4 b+3}{3-2 b-b^{2}}$

Algebra 2

Holliday, Luchin,… 1st Edition

Chapter 8, Problem 45

Instant Answer

Solved by Expert Elizabeth Xu

03/04/2020

Step 1

The numerator can be factored into $(b-1)(b-3)$ and the denominator can be factored into $-(b+3)(b-1)$. So, the expression becomes: \[\frac{(b-1)(b-3)}{-(b+3)(b-1)}\] Show more…

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Simplify each expression. $\frac{b^{2}-4 b+3}{3-2 b-b^{2}}$

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Key Concepts

Polynomial Factorization

This concept involves rewriting a polynomial as a product of its factors, typically simpler polynomials such as linear factors. Recognizing the factorable form of quadratics or other polynomial expressions is fundamental in simplifying algebraic expressions and solving equations.

Rational Expression Simplification

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying such expressions generally involves factoring both the numerator and the denominator and cancelling out any common factors, while also considering the restrictions on the variable that prevent division by zero.

Common Factor Cancellation

When the same factor appears in both the numerator and the denominator of a rational expression, it can be cancelled provided it is not equal to zero. This process reduces the expression to its simplest form and is a key step in simplification of rational expressions.

Handling Negative Signs

Identifying and appropriately handling negative signs is crucial when simplifying expressions. Negative signs may be factored out to reorganize terms in a polynomial, thus enabling clearer identification of common factors and ensuring the correctness of the simplification process.

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