Problem 42 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{9 b^{2}-16}{21 b+28} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{3b - 4}{7}\) with \(b \neq -\frac{4}{3}\).

Step 1: Factor the Numerator

The numerator of the expression is \(9b^2 - 16\), which is a difference of squares. Recall the formula for the difference of squares: \(a^2 - b^2 = (a - b)(a + b)\). Here, \(9b^2\) can be seen as \((3b)^2\) and 16 as \(4^2\). Therefore, the factoring is \((3b - 4)(3b + 4)\).

Step 2: Factor the Denominator

The denominator is \(21b + 28\). Look for common factors in each term. The greatest common factor (GCF) of 21 and 28 is 7. Factor out 7 from both terms: \(21b + 28 = 7(3b + 4)\).

Step 3: Simplify the Expression

The rational expression is now \(\frac{(3b - 4)(3b + 4)}{7(3b + 4)}\). Notice that \((3b + 4)\) appears in both the numerator and the denominator, so it can be cancelled as long as \(3b + 4 eq 0\) (if \(3b + 4 = 0\), the expression would be undefined). Cancel \((3b + 4)\) to simplify the expression to \(\frac{3b - 4}{7}\).

Step 4: Check for Restrictions

Since we cancelled \(3b + 4\) from both the numerator and denominator, make sure \(3b + 4eq 0\). Solving \(3b + 4 = 0\) gives \(b eq -\frac{4}{3}\). Thus, the simplified expression is valid for all \(b\) except \(-\frac{4}{3}\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial expression as a product of simpler expressions. It’s a crucial step in simplifying rational expressions. In our exercise, the numerator \(9b^2 - 16\) is a classic example.

Using the difference of squares, we broke it down using the formula \(a^2 - b^2 = (a - b)(a + b)\). Identifying that \(9b^2\) is \((3b)^2\) and \(16\) as \(4^2\), allows us to factor it as \((3b - 4)(3b + 4)\).

Tips for successful factoring:

• Always check if the expression is a perfect square or can be rewritten as a difference of squares.
• Search for common factors first, which can make the process simpler.
• Practice with a variety of expressions to recognize patterns quickly.

Simplifying Fractions

Simplifying fractions reduces them to their simplest form. For rational expressions, this means removing common factors in the numerator and denominator.

In our problem, after factoring, we recognize that \((3b + 4)\) is a common factor in both the numerator \((3b - 4)(3b + 4)\) and the denominator \(7(3b + 4)\). As long as \(3b + 4 eq 0\), this factor can be cancelled.

Points to remember while simplifying:

• Identify any restrictions first. Cancelled factors can sometimes equate to zero, making the expression undefined.
• Always re-evaluate whether additional simplification steps are possible after initial cancellation.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Simplifying them requires understanding various algebraic techniques.

Within the context of rational expressions, understanding how terms interact, such as common factors that appear in both numerator and denominator, is essential. Exploring and manipulating different algebraic expressions builds a strong foundation for understanding more complex mathematical concepts.

Some key tips:

• Recognize and apply algebraic identities, like the difference of squares or common factor techniques, for simplification.
• Be cautious of variable restrictions, ensuring you don't assume a value that could make the expression invalid or undefined.
• Practice on various algebraic expressions to build familiarity with how factors can be identified and cancelled.