Problem 28 - Algebra 2 and Trigonometry

Question

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{3-(b+1)}{4-b^{2}}\)

Step-by-Step Solution

Answer

Simplified expression: \(\frac{1}{2 + b}\); undefined at \(b = 2\) and \(b = -2\).

Step 1: Simplify the Numerator

The numerator of the expression is given as \(3 - (b+1)\). Distribute the negative sign inside the parenthesis: \(3 - b - 1\). Simplify by combining the constants: \(3 - 1 = 2\), resulting in \(2 - b\). Thus, the simplified numerator is:\(2 - b\).

Step 2: Simplify the Denominator Using Factoring

The denominator is \(4 - b^2\). Recognize this as a difference of squares: \((2^2) - (b^2) = (2 - b)(2 + b)\). So, the denominator is: \((2 - b)(2 + b)\).

Step 3: Simplify the Rational Expression

The simplified numerator is \(2 - b\) and the denominator is \((2 - b)(2 + b)\). Cancel the common factor \((2 - b)\) from the numerator and denominator. The simplified form of the expression is:\[\frac{1}{2 + b}\].

Step 4: Determine Values Where the Expression is Undefined

A rational expression is undefined when the denominator equals zero. Set the denominator of the original expression, \((2 - b)(2 + b)\), equal to zero: \((2 - b) = 0\) and \((2 + b) = 0\). Solving these gives \(b = 2\) and \(b = -2\). Therefore, the fraction is undefined at \(b = 2\) and \(b = -2\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental technique in algebra that allows us to transform a polynomial expression into simpler components called factors. Think of it as breaking down a complex structure into its basic building blocks.

• A polynomial is an expression like "ax² + bx + c" where the terms are combined using addition or subtraction.
• Factoring involves expressing the polynomial as a product of simpler polynomials.
• In many cases, these simpler polynomials are easier to work with, especially when simplifying fractions or solving equations.

When approaching a polynomial like the denominator in our exercise, which is "4 - b^2," recognizing it as a "difference of squares" is key. Not every polynomial will factor in this manner, so it's crucial to recognize different patterns or methods, like factoring by grouping or using the quadratic formula. To test if factoring is possible, see if there are two terms that are perfect squares separated by subtraction.

Difference of Squares

The difference of squares is a specific type of polynomial factoring that occurs when we subtract one perfect square from another. It fits the pattern:
\[ a^2 - b^2 = (a - b)(a + b) \]

• Recognize two terms that are squares: "2^2" and "b^2" in the exercise 4 - b².
• Apply the difference of squares pattern: set a = 2 and b = b . This gives us (2 - b)(2 + b).
• This factoring allows for simplification and solving of rational expressions.

Remember that the facility to identify such patterns quickly will greatly simplify many algebraic operations. Practice looking out for numbers and variables that can be expressed as squares, and you'll find solving these expressions becomes much more manageable.

Undefined Expressions

Understanding when expressions become undefined is critical when working with rational expressions. A rational expression is undefined for any value of the variable that makes the denominator zero, since division by zero is not allowed.

To determine these values:

• Set the denominator of a fraction equal to zero and solve for the variable.
• In the exercise, we have (2 - b)(2 + b) = 0. Solve each factor: (2 - b) = 0, which gives b = 2, and (2 + b) = 0, giving b = -2.
• The rational expression is undefined at these points, b = 2 and b = -2.

These values are crucial as they highlight points where the expression does not exist or behaves erratically. Always check for undefined values when dealing with fractions in algebra to prevent errors in calculations.