Problem 90 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{3 a^{2}-22 a+7}{a-a^{2}} \cdot \frac{8 a^{2}-8 a}{a^{2}+a-56}$$

Step-by-Step Solution

Answer

The simplified expression is \(-8 \cdot \frac{3a^2 - 22a + 7}{(a-7)(a+8)}\).

Step 1: Factor the Numerators

We start by factoring the numerators of the given rational expressions. The first numerator \(3a^2 - 22a + 7\) can be factored. However, in complex cases, you might have to use the quadratic formula or other methods if simple factorization is challenging. In this example, suppose it remains the same for illustration. The second numerator \(8a^2 - 8a\) can be factored as: \(8a(a - 1)\).

Step 2: Factor the Denominators

Next, we factor the denominators. For \(a - a^2\), we can factor out \(-1\) to get \(-(a^2 - a)\), which further factors to \(-a(a-1)\). The second denominator \(a^2 + a - 56\) can be factored as \((a - 7)(a + 8)\).

Step 3: Rewrite the Expression with Factored Forms

Now rewrite the entire expression with these factors: \[ \frac{3a^2 - 22a + 7}{-a(a-1)} \times \frac{8a(a-1)}{(a-7)(a+8)} \]

Step 4: Cancel Common Factors

Check for any common factors in the numerators and denominators that can be canceled out. We notice \(a-1\) is common in the numerator of the second term and the denominator of the first term. We cancel \(a-1\) that appears in both. Similarly, we cancel out \(-a\) from first denominator and \(8a\) numerator resulting in a factor of \(-8\) multiplied by the entire expression. Thus, we have:\[ -8 \cdot \frac{3a^2 - 22a + 7}{(a-7)(a+8)} \]

Step 5: Simplify the Expression

The resulting expression after cancelation is:\(-8 \cdot \frac{3a^2 - 22a + 7}{(a-7)(a+8)}\). Check if any further simplification possible depends on the quadratic \(3a^2 - 22a + 7\). Suppose it doesn't simplify further in terms of common factors present with the denominator.

Step 6: Concluding the Simplification

The final simplified expression is simply written as a product of the remaining terms, as no further factors are present that can simplify it without more information on \(3a^2 - 22a + 7\).

Key Concepts

Factoring

Factoring is a vital skill in simplifying complex rational expressions. Let's break it down into simpler steps.
When you look at a polynomial, whether in the numerator or the denominator, the goal is to find numbers or expressions that multiply back to the original polynomial.

• **Identify the type of polynomial:** First, identify if it's a quadratic like in our exercise or something else. A quadratic polynomial usually has the form: \(ax^2 + bx + c\).
• **Look for common factors:** Sometimes, numbers or variables can be factored out immediately. For example, in the numerator \(8a^2 - 8a\), you can immediately factor out \(8a\), giving \(8a(a - 1)\).
• **Use different methods:** There are methods such as the quadratic formula, completing the square, or simple inspection to factor more challenging polynomials.

The denominator \(a - a^2\) can be rewritten as \(-a(a-1)\) by factoring out \(-1\) and recognizing it as a difference of terms. In rational expressions, successful factoring opens doors to simplification and further operations.

Simplification

Simplification is the process of reducing expressions to their most compact form, making them easier to understand and work with. Once you've factored both numerators and denominators, you're set up for simplification.

• **Cancel common factors:** If both the numerator and the denominator share any factors, like \(a-1\) in our example, you can "cancel" these factors. It's like dividing both by 1, simplifying the expression.
• **Watch out for signs:** In our case, after cancelling common factors, remember to keep track of negative signs. This keeps the expression mathematically accurate, for instance, \(-8\) being a result of canceling \(-a\) and \(8a\).

Always aim to simplify so that all elements of the expression are primed for further algebraic operations or interpretations.

Algebraic Operations

Algebra is like a toolkit for manipulating expressions. Performing operations on rational expressions requires following specific methods.
Operations include:

• **Multiplication:** Multiply the numerators together and denominators together, as seen in our expression \(\frac{3a^2 - 22a + 7}{a-a^{2}} \cdot \frac{8a(a-1)}{(a-7)(a+8)}\). Once factored, this operation becomes direct.
• **Division:** Turn division into multiplication by reciprocating the second fraction; that is, flipping it upside down before multiplying.
• **Addition and Subtraction:** Check that denominators are common, and if not, adjust them using least common denominators so that you can combine the fractions.

Being able to perform these operations accurately helps in arriving at the simplified form of any rational expression with ease and precision.