Problem 81 - Algebra for College Students

Question

Perform the indicated operations. $$\frac{9 a-3}{1-9 a^{2}} \cdot \frac{9 a^{2}+6 a+1}{6}$$

Step-by-Step Solution

Answer

-\frac{3a + 1}{2}

Step 1: Factor the Denominators and Numerators

First, factor the expressions in the numerators and denominators. For the first fraction: The denominator is a difference of squares: \(1 - 9a^2 = (1 - 3a)(1 + 3a)\).For the second fraction: The numerator is a perfect square trinomial: \(9a^2 + 6a + 1 = (3a + 1)^2\).

Step 2: Rewrite the Fractions with Factored Form

Now rewrite the fractions using the factored forms:\[ \frac{9a - 3}{(1 - 3a)(1 + 3a)} \cdot \frac{(3a + 1)^2}{6} \]

Step 3: Simplify the First Fraction by Factoring

Factor out the common term in the numerator of the first fraction:\[ \frac{3(3a - 1)}{(1 - 3a)(1 + 3a)} \cdot \frac{(3a + 1)^2}{6} \]

Step 4: Simplify the Expression by Canceling Common Factors

Notice that \(3a - 1\) can be written as \(-(1 - 3a)\). Use this identity to facilitate canceling common factors:\[ \frac{3(-(1 - 3a))}{(1 - 3a)(1 + 3a)} \cdot \frac{(3a + 1)^2}{6} \]Simplify the first fraction:\[ \frac{-3}{(1 + 3a)} \cdot \frac{(3a + 1)^2}{6} \].Next, cancel out common terms in the fractions:\[ \frac{-3 \times (3a + 1)}{1 + 3a} \times \frac{(3a + 1)}{6} \].The term \((3a + 1)\) cancels out:\[ \frac{-3}{1 + 3a} \times \frac{(3a + 1)}{6} \].

Step 5: Final Simplification

Simplify the expression after canceling common terms:\[ \frac{-3 \times (3a + 1)}{6} = -\frac{3(3a + 1)}{6} \]Finally:\[ -\frac{3(3a + 1)}{6} = -\frac{3a + 1}{2} \].

Key Concepts

Understanding Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. They work much like numerical fractions but include variables.
In the exercise given, we work with algebraic fractions involving polynomials in the numerator and denominator.
To manipulate these fractions efficiently, we follow similar rules to regular fractions, such as finding common denominators, factoring, and canceling out common factors. This is crucial for simplifying the expressions and performing operations like addition, subtraction, multiplication, and division on them.

The Art of Factoring

Factoring is breaking down a complex expression into simpler terms (factors) that, when multiplied together, will give back the original expression.
In our exercise, we need to factor both the numerator and the denominator of the given algebraic fractions. Here’s a quick breakdown of our factors:
For the term \text{$$9a^2 + 6a + 1$$} we recognize this as a perfect square trinomial that can be factored as \text{$$(3a + 1)^2$$}.
For the term \text{$$1 - 9a^2$$}, we see a difference of squares which factors into \text{$$(1 - 3a)(1 + 3a)$$}. Understanding these factoring techniques is pivotal. It makes the process of simplifying and solving algebraic fractions manageable.

Simplification in Algebra

Simplification is about reducing an expression to its simplest form. This often requires factoring out common terms and canceling them appropriately.
In our exercise, after we factor the expressions, we look for common terms that can be canceled out. For instance:
The term \text{$$3a - 1$$} can be written as \text{$$-(1 - 3a)$$}, allowing it to match the factor in the denominator \text{$$(1 - 3a)$$}.
Matching and canceling these common factors simplifies the fraction significantly, digging down to the core of the expression efficiently. Mastery in simplification helps in reducing complex algebraic expressions to forms that are easier to handle and understand.

Understanding Polynomial Identities

Polynomial identities are equations that hold true for any value of the variables involved. They aid a lot in the factoring and simplification processes.
In this exercise, we leverage identities such as the difference of squares: \text($$a^2 - b^2 = (a - b)(a + b)$$). This insight allows us to quickly factor expressions, as shown with ($$1 - 9a^2$$).
Using polynomial identities, we simplify complex algebraic fractions swiftly by transforming them into more manageable terms. Recognizing these identities can tremendously speed up your problem-solving process, making tackling algebraic fractions less daunting.