Problem 44 - Algebra A Combined Function

Question

Multiply or divide as indicated. See Example 8. $$ \frac{b^{2}+2 b-3}{b^{2}+b-2} \cdot \frac{b^{2}-4}{b^{2}+6 b+8} $$

Step-by-Step Solution

Answer

\(\frac{(b + 3)(b - 2)}{b + 4}\)

Step 1: Factor the Numerator and Denominator of the First Fraction

We start by factoring the expressions in the first fraction. The numerator is \(b^2 + 2b - 3\) which factors into \((b + 3)(b - 1)\). The denominator \(b^2 + b - 2\) factors into \((b + 2)(b - 1)\). Therefore, the first fraction becomes \(\frac{(b + 3)(b - 1)}{(b + 2)(b - 1)}\).

Step 2: Factor the Numerator and Denominator of the Second Fraction

Next, we factor the expressions in the second fraction. The numerator \(b^2 - 4\) is a difference of squares and factors into \((b + 2)(b - 2)\). The denominator \(b^2 + 6b + 8\) factors into \((b + 2)(b + 4)\). Therefore, the second fraction becomes \(\frac{(b + 2)(b - 2)}{(b + 2)(b + 4)}\).

Step 3: Multiply the Fractions

To multiply the two fractions, we multiply the numerators together and the denominators together:\[\frac{(b + 3)(b - 1)(b + 2)(b - 2)}{(b + 2)(b - 1)(b + 2)(b + 4)}\]Cancel out the like terms present in the numerator and denominator. We can cancel \((b + 2)\) and \((b - 1)\) from both the numerator and the denominator.

Step 4: Simplify the Expression

After canceling common factors, the expression simplifies to:\[\frac{(b + 3)(b - 2)}{b + 4}\]This is as simplified as it can be.

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra that involves rewriting a polynomial as a product of its factors. This generally makes it easier to work with the expression, especially when performing operations like multiplication or division. When factoring, we look for common terms or patterns like the difference of squares or trinomials.

• For example, consider the polynomial \(b^2 + 2b - 3\). We can rewrite it as \((b + 3)(b - 1)\) by identifying two numbers that add to 2 (the coefficient of \(b\)) and multiply to -3 (the constant).
• Another example is \(b^2 - 4\), a classic difference of squares, which factors to \((b + 2)(b - 2)\).

Understanding how to factor polynomials is crucial as it allows us to simplify complex algebraic expressions, making subsequent operations like multiplying or simplifying far more manageable.

Multiplying Fractions

Multiplying fractions may sound intimidating, but it's an uncomplicated process once you understand the basics. When multiplying fractions, you simply multiply the numerators by each other and the denominators by each other. This principle also holds true when dealing with algebraic fractions.

• Start by factoring both the numerators and denominators of the fractions if they are polynomials. This step simplifies the multiplication process.
• For our example, we have \(\frac{(b + 3)(b - 1)}{(b + 2)(b - 1)} \cdot \frac{(b + 2)(b - 2)}{(b + 2)(b + 4)}\). To multiply, calculate: \((b + 3)(b - 1)(b + 2)(b - 2)\) for the numerator, and \((b + 2)(b - 1)(b + 2)(b + 4)\) for the denominator.

If common factors exist between the numerator and denominator, cancel them out before multiplying to ease the computation and simplify the final result.

Simplifying Expressions

Simplifying expressions is the process of making an expression as uncomplicated as possible. This often involves removing brackets, combining like terms, and reducing fractions. In the context of multiplying fractions, simplification often involves canceling common factors in the numerator and the denominator before performing the multiplication.

• Consider the expression \(\frac{(b + 3)(b - 1)(b + 2)(b - 2)}{(b + 2)(b - 1)(b + 2)(b + 4)}\). Before carrying out the multiplication, notice that \((b + 2)\) and \((b - 1)\) are common in both the numerator and the denominator.
• By canceling these terms, the expression reduces to \(\frac{(b + 3)(b - 2)}{b + 4}\). This considerably simplifies the process, leaving the expression in its simplest form.

Such simplifications are beneficial because they not only make calculations easier but also help you see the underlying relationships between variables more clearly.