Problem 65 - Introductory Algebra for College Students

Question

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{6 b^{2}-10 b}{16 b^{2}-48 b+27}+\frac{7 b^{2}-20 b}{16 b^{2}-48 b+27}-\frac{6 b-3 b^{2}}{16 b^{2}-48 b+27}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{5 b - 18 }{16 b - 48 + 27} \)

Step 1: Write out the expression

Write out the expression \( \frac{6 b^{2}-10 b}{16 b^{2}-48 b+27}+\frac{7 b^{2}-20 b}{16 b^{2}-48 b+27}-\frac{6 b-3 b^{2}}{16 b^{2}-48 b+27} \)

Step 2: Group like terms in the numerators

Since the denominators of each fractions are the same, you can group the like terms in the numerators together as: \( \frac{6 b^{2} + 7 b^{2} - 3 b^{2} -10 b - 20 b - 6 b}{16 b^{2} - 48 b + 27} \)

Step 3: Simplify the numerator

Combine the like terms in the numerator to get \( \frac{10 b^{2} - 36 b}{16 b^{2} - 48 b + 27} \)

Step 4: Factorize Numerator and Denominator

By taking out common factors, the expression simplifies to \( \frac{2b(5 b - 18)}{b(16 b - 48 + 27)} \)

Step 5: Cancelling out the common factor

Cancel out the common factor \(b\) in numerator and denominator to get \( \frac{5 b - 18 }{16 b - 48 + 27} \)

Key Concepts

Simplification

In algebra, simplification means rewriting an expression in a simpler or more concise form. It involves reducing fractions, combining like terms, and eliminating unnecessary complexity.
For the given problem, simplification starts by noting that all algebraic terms are fractions with the same denominator. So, we don't need to find a common denominator; we can directly add or subtract the numerators.
Here's how simplification works, step by step:

• Combine like terms in the numerators. For example, terms like \(6b^2\), \(7b^2\), and \(-3b^2\) can be added since they share the common variable with the same power.
• This results in the expression: \(\frac{10b^2 - 36b}{16b^2 - 48b + 27}\).
• The numerators and denominators are then examined further for possible factoring to simplify further.

Simplification reduces fractions to their simplest form, making them easier to understand and work with in further mathematical operations without altering the expression's value.

Factoring

Factoring is the process of breaking down an algebraic expression into simpler terms or 'factors' that, when multiplied together, give back the original expression.
In our exercise:

• The expression in its simplified form from the previous step is \(\frac{10b^2 - 36b}{16b^2 - 48b + 27}\).
• We factor the numerator to \(2b(5b - 18)\), by taking out the greatest common factor, which in this case is \(2b\).
• The denominator can be part of a more complex factoring process, possibly involving trial and error, but we take out \(b\) as a factor, simplifying it to \(b(16b - 48 + 27)\).

Factoring is vital for reducing rational expressions as it reveals common factors that can be canceled out, simplifying the expression further. It requires practice, but using common factor tricks and pattern recognition significantly aids factoring.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. They resemble numerical fractions, but with variables.
Here's why rational expressions matter:

• They play a crucial role in algebra, calculus, and many applied mathematical fields.
• Understanding them involves operations like addition, subtraction, multiplication, and division, similar to numeric fractions, but requiring algebraic manipulation.

In the exercise, the entire operation revolved around dealing with rational expressions. Since the denominators were the same, operations were simplified to focusing on manipulating the numerators.
Ultimately, after factoring, we obtained \(\frac{5b - 18}{16b - 48 + 27}\). Rational expressions require a good grasp of both numerators and denominators' behavior to simplify and solve correctly. With consistent practice, working with rational expressions becomes second nature.