Problem 96 - College Algebra

Question

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations. $$g(x)=\frac{3 x+7}{x+2}$$

Step-by-Step Solution

Answer

The function \(g(x)\) in the form of quotient + remainder / divisor is given by \(g(x) = 3 + \frac{1}{x + 2}\)

Step 1: Perform Polynomial Long Division

Firstly, perform long division on the equation \(g(x) = \frac{3x + 7}{x + 2}\). Divide 3x + 7 by x + 2. When you divide 3x by x, you get a quotient of 3. Multiply x + 2 by 3 and subtract the result from 3x + 7, then you get a remainder of 1.

Step 2: Write in Quotient and Remainder Form

After the long division, you will find that the quotient is 3 and the remainder is 1. So, we write the division in terms of quotient and remainder: \(g(x) = 3 + \frac{1}{x + 2}\)

Step 3: Interpret the Result

Compare the resultant equation with original equation. The first term 3 in the resultant equation represents the quotient obtained and the term \(\frac{1}{x + 2}\) is the remainder divided by divisor. This form of representation is helpful in understanding the transformation of the original function.

Key Concepts

Quotient and Remainder

When we talk about polynomial long division, the concepts of 'quotient and remainder' are fundamental. Much like traditional numerical division, where you divide a number to find how many times the divisor fits into the dividend and what’s left over is the remainder, polynomial long division follows a similar pattern but with algebraic expressions.

In our exercise, for instance, we divided the polynomial function’s numerator by its denominator. We found that when dividing the term '3x' by 'x', the result, or quotient, is simply '3'. Once we subtract the product of the quotient and the divisor from our initial polynomial, we're left with a remainder. Here, that remainder is '1'. Instead of leaving our function as a fraction, we're able to express it in a more digestible form: the original polynomial equals the quotient plus the remainder over the divisor; or in our case, this is expressed as:\(g(x) = 3 + \frac{1}{x + 2}\).

Understanding how to identify and write the quotient and remainder helps significantly in simplifying polynomial expressions and in further analysis of their properties.

Function Transformation

Function transformation is a critical concept in algebra that involves changing the position or size of a graph without altering its shape. This can include shifting it up, down, left, or right, stretching or compressing it, and even reflecting it across an axis. These transformations are based on modifications to the function's formula.

In our example of rewriting the function \(g(x) = \frac{3x + 7}{x + 2}\), the quotient reveals a horizontal shift in the graph of the rational function. The '3' in the rewritten function \(g(x) = 3 + \frac{1}{x + 2}\) indicates that every point on the graph has been moved three units up on the y-axis. The remainder over the divisor, or \(\frac{1}{x + 2}\), implies a horizontal shift to the left by 2 units and a vertical asymptote at \(x = -2\).

Recognizing and understanding these shifts and their implications on the graph can be incredibly useful, especially when graphing functions by hand or predicting the behavior of a function based on its equation.

Rational Functions Algebra

Algebra involving rational functions deals with expressions describing a ratio of two polynomials, much like the one in our exercise, \(g(x) = \frac{3x + 7}{x + 2}\). The algebraic operations on these functions can include addition, subtraction, multiplication, division, and simplification, which often involve finding common denominators or factoring polynomials.

An important aspect of rational functions is their behavior: they can approach infinity or negative infinity (vertical asymptotes), level off to a horizontal value (horizontal asymptotes), or have holes where they are undefined due to common factors in the numerator and denominator.

The polynomial long division, as demonstrated in our exercise, simplifies the given rational function into a more manageable form, which then aids in the analysis of these behaviors, plotting graphs, and solving equations involving rational functions. It's these transformative algebraic manipulations that allow us to deeply understand the nature of complex rational expressions.