Problem 47 - Algebra A Combined Function

Question

Multiply or divide as indicated. See Example 8. $$ \frac{8 n^{2}-18}{2 n^{2}-5 n+3} \div \frac{6 n^{2}+7 n-3}{n^{2}-9 n+8} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{2(n-8)}{3n-1} \).

Step 1: Rewrite the Division as Multiplication

To divide by a fraction, you can multiply by its reciprocal. So, we rewrite: \[ \frac{8n^{2} - 18}{2n^{2} - 5n + 3} \div \frac{6n^{2} + 7n - 3}{n^{2} - 9n + 8} \] as \[ \frac{8n^{2} - 18}{2n^{2} - 5n + 3} \times \frac{n^{2} - 9n + 8}{6n^{2} + 7n - 3} \].

Step 2: Factor All Polynomials

The next step is to factor each polynomial if possible. - For \(8n^{2} - 18\), factor out the greatest common factor: \(2(4n^{2} - 9) = 2(2n - 3)(2n + 3)\).- For \(2n^{2} - 5n + 3\), factor as \((2n - 3)(n - 1)\).- For \(6n^{2} + 7n - 3\), factor as \((3n - 1)(2n + 3)\).- For \(n^{2} - 9n + 8\), factor as \((n - 1)(n - 8)\).

Step 3: Substitute Factored Terms

Substitute the factored forms into the expression:\[ \frac{2(2n - 3)(2n + 3)}{(2n - 3)(n - 1)} \times \frac{(n - 1)(n - 8)}{(3n - 1)(2n + 3)} \].

Step 4: Cancel Common Factors

Cancel the common factors across numerators and denominators:- \((2n - 3)\) appears in both the numerator and denominator.- \((2n + 3)\) also appears in both the numerator and denominator.- \((n - 1)\) appears in both the numerator and denominator.After cancelling, we are left with:\[ \frac{2(n - 8)}{3n - 1} \].

Step 5: Simplify the Expression

With all common factors cancelled, the expression simplifies to:\[ \frac{2(n - 8)}{3n - 1} \].This is the final simplified form of the original division problem.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that simplifies expressions, especially when dealing with rational expressions like division or multiplication of fractions. The idea is to break down complex polynomial expressions into products of their simpler factors. This is similar to splitting a large number into a multiplication of its prime components, but in terms of variables and coefficients.
For instance, consider the polynomial \(8n^2 - 18\). Here, we start by identifying any common factors of all terms, which is 2 in this case. Factoring it out, the expression becomes \(2(4n^2 - 9)\). The remaining quadratic \(4n^2 - 9\) fits the difference of squares pattern, \(a^2 - b^2 = (a - b)(a + b)\), so it further factors to \((2n - 3)(2n + 3)\).
Breaking polynomials down this way makes them easier to manage and simplifies the process of division by highlighting common factors across expressions.

Rational Expressions

Rational expressions are fractions that have polynomials in their numerators and denominators. Dealing with them often involves operations similar to those with numeric fractions but requires managing the variable terms and ensuring the expressions are correctly simplified.
When dividing rational expressions, follow these steps:

• Rewrite the division as multiplication by the reciprocal of the second fraction.
• Factor the polynomials completely to unveil any commonalities across the fractions.
• Substitute these factored forms back into the expression for simplification.

For example, the division of \(\frac{8n^2 - 18}{2n^2 - 5n + 3}\) by \(\frac{6n^2 + 7n - 3}{n^2 - 9n + 8}\) is converted to a multiplication setup and simplified further with factoring. This process underscores the importance of changing how you view the division to help clarify and unify the expression for simplification.

Cancelling Common Factors

Once rational expressions are fully factored, cancelling common factors becomes a straightforward task. This operation is vital to simplify the expression to its most reduced form, ensuring clarity and correctness.
In our example, after factoring, the expression becomes:\[\frac{2(2n - 3)(2n + 3)}{(2n - 3)(n - 1)} \times \frac{(n - 1)(n - 8)}{(3n - 1)(2n + 3)}\]Spot the repeating factors in the numerators and denominators, such as \((2n - 3)\), \((2n + 3)\), and \((n - 1)\). By cancelling these common factors, you simplify the expression directly to:\[\frac{2(n - 8)}{3n - 1}\]This process highlights how reducing complex fractions hinges on eliminating shared terms to clear up any redundancy, thus simplifying it to its essence. This makes the expression easier to understand and apply in further calculations.