Problem 78 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{3 a+6}{4 a-24} \cdot \frac{6-a}{3 a+15} $$

Step-by-Step Solution

Answer

Based on the given solution, the simplified product of the two fractions is: $$ \frac{-a^2 + 4a + 12}{4(a-6)(a+5)} $$

Step 1: Simplify each expression separately

Let's first simplify each fraction separately by factoring out common terms of numerators and denominators. For the first fraction, we can factor out a 3 from the numerator, and a 4 from the denominator. For the second fraction, we can factor out a 3 from the denominator. $$ \frac{3(a+2)}{4(a-6)} \cdot \frac{6-a}{3(a+5)} $$

Step 2: Look for common factors to simplify further

Before multiplying the fractions, we need to look for any common factors in the numerators and denominators that can be canceled out. In this case, we have a common factor of 3. We can simplify the fractions by canceling out the common factor: $$ \frac{a+2}{4(a-6)} \cdot \frac{6-a}{(a+5)} $$

Step 3: Multiply the simplified fractions

Now that we have simplified both fractions, we can multiply them. To multiply fractions, we multiply the numerators together and the denominators together. The numerator of the product will be \((a+2)(6-a)\) and the denominator will be \(4(a-6)(a+5)\). Thus we have: $$ \frac{(a+2)(6-a)}{4(a-6)(a+5)} $$

Step 4: Distribute the terms in the numerator

We will multiply the terms \((a+2)\) and \((6-a)\) in the numerator by distributing the terms. Following the FOIL method, we'll multiply First terms, Outer terms, Inner terms, and finally the Last terms, and then sum them up: $$ (a+2)(6-a) = a(6) - a^2 + 2(6) - 2(a) = -a^2 + 4a + 12 $$ So our final expression is: $$ \frac{-a^2 + 4a + 12}{4(a-6)(a+5)} $$

Key Concepts

Multiplying Fractions

Multiplying fractions involve an easy two-step process. The first step is to multiply the numerators, and the second step is to do the same with the denominators. Consider two fractions that need multiplication: \( \frac{a}{b} \times \frac{c}{d} \). To multiply these fractions, follow the simple formula:

• Multiply the numerators: The new numerator will be \( a \times c \).
• Multiply the denominators: The new denominator will be \( b \times d \).

Multiplying fractions is straightforward because it eliminates the need for finding a common denominator, a step necessary only when adding or subtracting fractions. In the given exercise, the multiplication was done with simplified fractions, making the process much smoother. Always remember that you can simplify the expression before or after multiplying the fractions, and simplification usually makes multiplication easier.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This means making the expression as concise as possible without changing its value. This is an important step when working with algebraic fractions. The simplification process usually includes:

• Identifying and factoring out common terms: Check the numerator and denominator for any common factors you can cancel out.
• Reducing the expression: Dividing out common terms to simplify fractions.

In our exercise, each fraction was simplified by factoring common terms first. For example, the numerator \( 3a+6 \) was factored as \( 3(a+2) \), making it simpler to identify common factors. Cutting through complex layers in algebraic expressions helps in finding more straightforward solutions.

Factoring in Algebra

Factoring in algebra involves expressing an algebraic expression as a product of its factors. It's like taking apart a mathematical equation to see the pieces that make it whole. Factoring can greatly simplify expressions and involves looking for common factors or applying specific techniques, such as:

• Recognizing common terms: A factor such as a number or variable that appears in every term of an expression.
• Using standard factoring techniques: Such as grouping, difference of squares, or special factoring formulas like \( a^2 - b^2 = (a-b)(a+b) \).

For example, in the exercise, the expression \( 3a+6 \) was factored to \( 3(a+2) \). This simplifies seeking common factors across the fraction, allowing for easier cancellation and simplification of the expression.

FOIL Method

The FOIL method is a systematic way to multiply two binomials. It's an acronym representing the order in which you multiply the terms:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms in the product.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms in each of the binomials.

Adding the results gives the expanded form of the expression.
In the exercise example, the binomials \( (a+2) \) and \( (6-a) \) were multiplied using FOIL. Each component was carefully multiplied and subsequently added together. The result was the expression \( -a^2 + 4a + 12 \), which became the simplified numerator of the product.
While it might seem intimidating at first, with a bit of practice, the FOIL method becomes a useful tool for working with polynomials.