Problem 33 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 3. $$ \frac{3 a^{2}+7 a b+2 b^{2}}{a^{2}+2 a b} \cdot \frac{a^{2}-a b}{3 a^{2}+a b} $$

Step-by-Step Solution

Answer

The expression simplifies to \(a-b\).

Step 1: Factor the first numerator

The first numerator is \(3a^2 + 7ab + 2b^2\). To factor it, look for two numbers that multiply to \(3 \times 2 = 6\) and add to 7. These numbers are 6 and 1. Thus, rewrite the middle term: \(3a^2 + 6ab + 1ab + 2b^2\). Group and factor each pair: \((3a^2 + 6ab) + (ab + 2b^2)\) which simplifies to \(3a(a + 2b) + b(a + 2b)\). Hence, the factor is \((3a + b)(a + 2b)\).

Step 2: Factor the first denominator

The first denominator is \(a^2 + 2ab\). Factor out the greatest common factor, \(a\), to get \(a(a + 2b)\).

Step 3: Factor the second numerator

The second numerator is \(a^2 - ab\). Factor out the greatest common factor, \(a\), to get \(a(a - b)\).

Step 4: Factor the second denominator

The second denominator is \(3a^2 + ab\). Factor out the greatest common factor, \(a\), to get \(a(3a + b)\).

Step 5: Set up the entire expression

The expression is now:\[ \frac{(3a + b)(a + 2b)}{a(a + 2b)} \cdot \frac{a(a - b)}{a(3a + b)} \]

Step 6: Cancel common factors

Cancel \(a + 2b)\) from the first fraction and \(a\) from both fractions. This simplifies the expression to \[\frac{3a + b}{1} \cdot \frac{a - b}{3a + b}\]. Additionally, cancel \(3a + b\) in the numerator with the denominator giving \[a - b\].

Step 7: Simplify the final expression

The expression simplifies to \(a-b\). Thus, the original expression simplifies entirely to \(a-b\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental technique in intermediate algebra that involves breaking down a polynomial into the product of its simpler factors. This is similar to finding the prime factorization of a number, but for expressions containing variables. In our example, the first polynomial in the numerator is \(3a^2 + 7ab + 2b^2\). To factor this, we need to find two numbers that multiply to the product of the leading coefficient and the constant term, \(3 \times 2 = 6\), and add up to the middle term's coefficient, 7. This leads us to the numbers 6 and 1.

Using these numbers, we rewrite the polynomial as \(3a^2 + 6ab + 1ab + 2b^2\) and group the terms: \((3a^2 + 6ab) + (ab + 2b^2)\). By factoring each group, we obtain \(3a(a + 2b) + b(a + 2b)\). Notice that \(a + 2b\) is a common factor, allowing us to write the factored form as \((3a + b)(a + 2b)\).

Understanding factoring is crucial as it simplifies expressions and is foundational for solving equations and rational expressions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions helps to make equations easier to work with by reducing them to their simplest form. For our exercise, once the polynomial expressions have been factored, we then simplify the overall expression by canceling out common factors.

After rewriting our expression in factored form:

• \(\frac{(3a + b)(a + 2b)}{a(a + 2b)} \cdot \frac{a(a - b)}{a(3a + b)}\)

we perform simplification. We first note the common factor \(a + 2b\) in the numerator and denominator of the first fraction, which can be canceled. Similarly, the factor \(a\) is common in both the numerators, allowing further cancellation. Finally, by evaluating the resulting product, we identify \((3a + b)\) as a common factor again. This complete cancellation process leaves us with the simplified answer of \(a-b\).

Simplifying expressions by canceling out these common factors ensures that we only deal with essential terms, making it easier to understand and solve the problem.

Rational Expressions

Rational expressions are quotients of polynomials, much like fractions in basic arithmetic. They require a firm understanding of both factoring and simplification to manipulate effectively. In our example, you’re working with two rational expressions multiplied together:

• \(\frac{3a^2 + 7ab + 2b^2}{a^2 + 2ab} \cdot \frac{a^2 - ab}{3a^2 + ab}\)

Factoring each component as shown in the previous steps allows you to set up for simplification. The key is identifying and canceling out like terms in the numerators and denominators.

In the context of rational expressions, it’s crucial to recognize that any simplification must preserve the expression’s original value except where undefined (such as division by zero). By canceling common factors, we avoid unnecessary complexity and reach our solution, which is \(a-b\). This simplification aids in revealing the essence of the expressions involved and allows us to understand their behavior more clearly.