Problem 45 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{2 a b+2 b y+3 a+3 y}{2 b^{2}-7 b-15}$$

Step-by-Step Solution

Answer

\(\frac{a + y}{b - 5}\)

Step 1: Factor the numerator

Factor out the common term in the numerator. The numerator is: \(2ab + 2by + 3a + 3y\). We can factor by grouping: \(2b(a + y) + 3(a + y)\). Now factor out the common factor \((a + y)\): \((a + y)(2b + 3)\).

Step 2: Factor the denominator

Factor the quadratic expression in the denominator: \(2b^2 - 7b - 15\). To do this, find two numbers that multiply to \(2 \times (-15) = -30\) and add to \(-7\). These numbers are \(-10\) and \(3\). Rewrite the middle term using these numbers: \(2b^2 - 10b + 3b - 15\). Now, factor by grouping: \(2b(b - 5) + 3(b - 5)\). Factor out the common factor \((b - 5)\): \((b - 5)(2b + 3)\).

Step 3: Simplify the expression

The numerator and the denominator both contain the common factor \((2b + 3)\). Cancel out this common factor: \(\frac{(a + y)(2b + 3)}{(b - 5)(2b + 3)} = \frac{a + y}{b - 5}\).

Key Concepts

Factoring Polynomials

To reduce rational expressions, a key technique is factoring polynomials. Factoring involves breaking down a polynomial into its simplest parts or 'factors'. This is like rewriting a number like 12 as the product of its factors: 3 and 4. For polynomials, the goal is to express them as a product of simpler polynomials. For example, consider the polynomial in our exercise's numerator: \(2ab + 2by + 3a + 3y\). By grouping terms, we can factor it as \((a + y)(2b + 3)\). It's essential to master this skill because it allows for simplifying complex expressions and solving polynomial equations more easily.

Simplifying Expressions

Simplifying expressions means making them as simple as possible without changing their value. After factoring polynomials in a rational expression, the next step is to identify and cancel out common factors from the numerator and denominator. This acts like reducing a fraction: just as \(\frac{6}{9}\) simplifies to \(\frac{2}{3}\) by cancelling the common factor of 3, we can simplify rational expressions by eliminating shared polynomial factors. In our example, once we factorize both the numerator and the denominator, we find they both share the factor \(2b + 3\). By cancelling this common factor, we reduce the expression to its lowest terms: \(\frac{a + y}{b - 5}\). Simplifying in this way makes expressions easier to work with and understand.

Quadratic Equations

Quadratic equations often appear in rational expressions. A quadratic equation is any equation that can be written in the form \(ax^2 + bx + c = 0\). The denominator in our exercise, \(2b^2 - 7b - 15\), is a quadratic expression. To simplify it, we 'factor' it into products of simpler polynomials. This can involve finding two numbers that multiply to give the product of \(a\) and \(c\), and add up to \(b\). In our example, the quadratic \(2b^2 - 7b - 15\) factors into \((b - 5)(2b + 3)\). By rewriting the quadratic in this factored form, it becomes easier to cancel out and simplify.

Common Factors

Common factors are values or expressions that divide evenly into two or more terms. Identifying and cancelling them is a key step in simplifying expressions. In our exercise, after factoring both the numerator and the denominator, we find they both share the factor \(2b + 3\). By recognizing this common factor, we can cancel it from both the numerator and the denominator. This step is crucial because it reduces the rational expression to its simplest form. Regular practice in spotting and cancelling common factors will greatly aid in simplifying various algebraic expressions.