Problem 87 - Intermediate Algebra

Question

a. \(\frac{16 n^{2}-16 n-5}{4 n}\) b. \(\frac{16 n^{2}-16 n-5}{4 n+1}\)

Step-by-Step Solution

Answer

a) \(4n - 4 - \frac{5}{4n}\) b) \(4n - 5 - \frac{10}{4n + 1}\)

Step 1: Simplification of Numerator for Part (a)

Let's start with reviewing the expression in the numerator of part (a), which is \(16n^2 - 16n - 5\). This term cannot be factored further with integer values so we will directly divide it by the denominator in part (a).

Step 2: Simplify Part (a)

Divide the entire expression \(16n^2 - 16n - 5\) by \(4n\).\[\frac{16n^2 - 16n - 5}{4n} = \frac{16n^2}{4n} - \frac{16n}{4n} - \frac{5}{4n}\]This simplifies to \[4n - 4 - \frac{5}{4n}\].

Step 3: Simplification of Numerator for Part (b)

Now, let's consider the numerator for part (b), which is the same expression: \(16n^2 - 16n - 5\). Like in part (a), this expression cannot be factored with integer values directly.

Step 4: Division for Part (b)

For part (b), divide the entire expression \(16n^2 - 16n - 5\) by \(4n + 1\). This involves polynomial long division because \(4n + 1\) does not evenly divide the polynomial. The quotient will be found and any remainder expressed as a fraction.Using polynomial long division:Quotient: \(4n - 5\)Remainder: \(-10\)So, the division becomes:\[4n - 5 - \frac{10}{4n + 1}\].

Key Concepts

Simplification

Simplification is a key step in any mathematical problem-solving process. It involves reducing an expression to its simplest form. In this context, simplifying a polynomial expression by dividing it can make calculations more straightforward. For Part (a) of the original exercise, we are given the expression \(\frac{16n^2 - 16n - 5}{4n}\). Although the numerator \(16n^2 - 16n - 5\) cannot be factored with integer values, we can break it down by dividing each term individually by the denominator \(4n\).
This leads to:

• \(\frac{16n^2}{4n} = 4n\)
• \(-\frac{16n}{4n} = -4\)
• \(-\frac{5}{4n}\) remains as it is because it's already in a simplified fractional form.

Putting these simplified parts together gives us \(4n - 4 - \frac{5}{4n}\). This simplifies the task, making further computations easier.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. They are similar to rational numbers in arithmetic, but they contain variables. It's important to simplify rational expressions wherever possible. This simplification can involve factoring, canceling common terms, or using algebraic manipulation.
In the original problem, we're dealing with rational expressions like \(\frac{16n^2 - 16n - 5}{4n+1}\). Here, neither the numerator nor the denominator can be reduced through simple factoring. Nevertheless, each can be handled using polynomial division to convert the expression into a simpler form, which is easier to work with or evaluate for specific values of \(n\). Manipulating rational expressions requires respecting the domain, meaning any value that makes the denominator zero must be considered, as division by zero is undefined. In our example, \(n\) must not be zero in Part (a) and not \(-\frac{1}{4}\) in Part (b). This ensures the expressions remain valid.

Polynomial Long Division

Polynomial long division is a method used for dividing a polynomial by another polynomial of lower degree. It's similar to the long division process used with numbers and helps us find both the quotient and the remainder.
For Part (b) of the exercise, the expression \(\frac{16n^2 - 16n - 5}{4n + 1}\) requires polynomial long division. Here’s a quick guide through the division process: - We start by dividing the leading term of the numerator \(16n^2\) by the leading term of the denominator \(4n\), giving us \(4n\).- Multiply \(4n\) by \(4n + 1\), then subtract the result from the original numerator to find the new expression.- Repeat the process with the new expression. The division reveals the quotient \(4n - 5\) and a remainder of \(-10\).These steps give us the answer \(4n - 5 - \frac{10}{4n + 1}\). Polynomial long division is critical when simplifying expressions that involve rational polynomials, especially when direct factoring isn't feasible.