Problem 21 - Algebra A Combined Function

Question

Find each quotient and simplify. See Examples 4 through 7. $$ \frac{(x-6)(x+4)}{4 x} \div \frac{2 x-12}{8 x^{2}} $$

Step-by-Step Solution

Answer

The simplified quotient is \( x^2 + 4x \).

Step 1: Understand the Division of Fractions

When dealing with division of fractions, recall that dividing by a fraction is equivalent to multiplying by its reciprocal. We will convert the division problem into a multiplication problem by taking the reciprocal of the second fraction.

Step 2: Rewrite the Expression

Rewrite the given problem from \[ \frac{(x-6)(x+4)}{4x} \div \frac{2x-12}{8x^2} \]to a multiplication problem by taking the reciprocal of the second fraction:\[ \frac{(x-6)(x+4)}{4x} \times \frac{8x^2}{2x-12} \] .

Step 3: Factor Common Terms

Identify and factor out any common terms in the numerators and denominators. The term \(2x - 12\) can be factored as \(2(x-6)\). So, the multiplication expression becomes:\[ \frac{(x-6)(x+4)}{4x} \times \frac{8x^2}{2(x-6)} \].

Step 4: Cancel Common Factors

Cancel the common factor \((x-6)\) in the numerator and the denominator.\[ \frac{(x+4)}{4x} \times \frac{8x^2}{2} \]. Then, simplify further by reducing the fraction \(\frac{8x^2}{2}\) to \(4x^2\), giving:\[ \frac{(x+4)}{4x} \times 4x^2 \].

Step 5: Multiply and Simplify

Multiply the remaining fractions:\[ \frac{(x+4)}{4x} \times 4x^2 = (x+4) \times x \].Cancel the \(x\) from numerator and denominator, yielding \[ x(x+4) = x^2 + 4x \].

Step 6: Conclusion

The quotient, when simplified, gives \( x^2 + 4x \).

Key Concepts

Polynomial Division

In algebra, dividing polynomials often requires handling algebraic fractions. The trick lies in understanding that dividing by a fraction is akin to multiplying by its reciprocal. This rule simplifies the process drastically.

When you encounter a division of polynomials, such as \( \frac{(x-6)(x+4)}{4x} \div \frac{2x-12}{8x^2} \), you should first convert it into a multiplication problem. This is achieved by taking the reciprocal of the second polynomial fraction, thus turning it into a multiplication operation between two fractions. This approach is straightforward and reduces the complexity of the original division problem.

• Convert division to multiplication using reciprocals.
• The division sign transforms the problem into a multiplication of the first expression by the inverse of the second.

After rewriting the expression, the next step involves simplifying and factoring, which often makes the expression much easier to handle.

Simplifying Expressions

Simplifying algebraic expressions involves reducing them into their simplest form. This process strips away unnecessary complexities and makes computations more manageable.

In our given expression, once you've rewritten the division of polynomials as a multiplication problem, you begin simplifying by factoring common terms. Specifically, you need to break down expressions where possible, like turning \((2x-12)\) into \(2(x-6)\). Identifying and canceling common elements both in the numerator and the denominator is critical here.

• Look for terms that can factor out from the expressions.
• Cancel out identical terms from the numerator and denominator to simplify.

Finally, after canceling common terms, you should multiply the remaining expressions together and simplify further if possible. This straightforward aspect of algebra can often be the key to arriving at neat, easily understandable solutions.

Factoring Polynomials

Factoring polynomials is a fundamental skill that lets you break down complex polynomials into simpler components. These components are products of simpler, non-divisible polynomials. It is especially useful in simplifying expressions and solving equations involving polynomials.

Take the expression \(2x-12\) from our example. You can factor it by extracting a common factor, \(2\), resulting in \(2(x-6)\). Factoring such expressions permits canceling similar terms in complex algebraic fractions, thus simplifying the entire polynomial division process.

• Identify the greatest common factor in polynomial expressions.
• Decompose complex polynomials into smaller, more manageable components.

The ultimate goal is to make the polynomial manageable, allowing further operations like multiplication or further simplification to be executed effortlessly.