Problem 34 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 3. $$ \frac{a^{2}+3 a b+2 b^{2}}{a^{2}-3 a b-4 b^{2}} \cdot \frac{a^{2}-4 a b}{a b^{2}+2 b^{3}} $$

Step-by-Step Solution

Answer

The simplified result of the expression is \( \frac{a}{b^2} \).

Step 1: Factor the Numerators

First, let's factor the numerator of the first fraction, which is \( a^2 + 3ab + 2b^2 \). This can be factored into \((a + b)(a + 2b)\). The numerator of the second fraction, \( a^2 - 4ab \), can be factored by taking out the greatest common factor \(a\), resulting in \(a(a - 4b)\).

Step 2: Factor the Denominators

Next, let's factor the denominator of the first fraction, \( a^2 - 3ab - 4b^2 \). This can be factored as \((a - 4b)(a + b)\). For the denominator of the second fraction, \( ab^2 + 2b^3 \), factor out the greatest common factor \(b^2\), giving \(b^2(a + 2b)\).

Step 3: Set Up the Expression for Multiplication

After factoring, we rewrite the entire expression using the factored forms: \[ \frac{(a + b)(a + 2b)}{(a - 4b)(a + b)} \cdot \frac{a(a - 4b)}{b^2(a + 2b)} \].

Step 4: Cancel Common Factors

Now, let's cancel out common factors found in the numerators and denominators:- \(a + b\) is present in both the numerator and denominator of the first fraction.- \(a + 2b\) is present in both the numerator and the denominator of the second fraction.- \(a - 4b\) is present in both the numerator and the denominator after multiplication.After cancelling, we are left with: \[ \frac{a}{b^2} \].

Step 5: Simplify the Expression

The expression \( \frac{a}{b^2} \) is already simplified, so no further steps are necessary.

Key Concepts

Factoring Expressions

Factoring expressions is a fundamental skill in algebra that involves breaking down a complicated expression into simpler components called factors. Each factor is a building block of the original expression. To make the task easier, look for patterns or common terms in the expression that can be grouped or simplified.

• Begin by identifying if there are any common factors across the terms in the expression.
• Then, see if the expression can be represented using recognizable patterns like difference of squares or trinomial forms.

In the exercise, the numerator of the first fraction has a trinomial of the form \( a^2 + 3ab + 2b^2 \). By examining the coefficients and variables, you can factor it as \( (a + b)(a + 2b) \). This means that the expression can be expressed as the product of two binomials, making it easier to work with.

Next, for the numerator of \( a^2 - 4ab \), recognizing a common factor \(a\) allows us to factor it as \(a(a - 4b)\). Mastering factoring techniques simplifies complex expressions and paves the way for solving equations or transforming expressions into equivalent forms.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Just like numerical fractions, performing operations requires careful handling of these expressions. It is essential to understand their structure to simplify or manipulate them effectively.

• The expression \( \frac{a^{2}+3 a b+2 b^{2}}{a^{2}-3 a b-4 b^{2}} \) is a rational expression because both the numerator and the denominator are polynomials.
• Operations like addition, subtraction, multiplication, and division are possible, but they often require factoring and simplification techniques to manage the terms efficiently.

When working with rational expressions, ensuring that no division by zero occurs is crucial, implying the need for cautious simplification especially when dealing with inequalities.

By factoring polynomials, we can often identify common factors that can be canceled out, making rational expressions appear less convoluted and more manageable. This method is a necessity during multiplication and division of these expressions.

Simplifying Algebraic Fractions

Simplifying algebraic fractions is similar to simplifying numerical fractions, but it involves polynomials. When you simplify an algebraic fraction, the goal is to reduce it to its simplest form by canceling out common factors in the numerator and the denominator.

• First, factor both the numerator and the denominator completely.
• Next, identify and cancel any common factors present in both, thus reducing the fraction.

In our exercise, the expression \[ \frac{(a + b)(a + 2b)}{(a - 4b)(a + b)} \cdot \frac{a(a - 4b)}{b^2(a + 2b)} \] involves cancelling common factors across the fractions. After cancellation, you get a more straightforward expression \( \frac{a}{b^2} \).

This final expression shows that simplifying can significantly cut down the complexity of algebraic frameworks, allowing for clearer interpretation and solution of problems. It is crucial to ensure that no variable in the denominator equates to zero, preserving the mathematical integrity of the expression.