91Ó°ÊÓ

Rational functions are expressions that take the form of one polynomial divided by another polynomial. In our exercise, we have the fraction \(\frac{P(x)}{d(x)}\), where \(P(x) = 5x^3 + 2x^2 - 3x\) is the numerator and \(d(x) = x - 3\) is the denominator. To simplify or analyze rational functions, we often rewrite them in a different form. This can make operations such as integration or finding limits easier.

In this exercise, rewriting \(\frac{P(x)}{d(x)}\) in the form \(Q(x) + \frac{r(x)}{d(x)}\) helps break down complex polynomial fractions into more manageable parts. Here, \(Q(x)\) is the quotient obtained through division, and \(r(x)\) is the remainder. Understanding this format is key when applying further mathematical operations on rational functions.

Long division with polynomials works similarly to long division with numbers. The process helps break down a polynomial into simpler components. Let's look at how it's done.

• First, divide the leading term of the numerator by the leading term of the denominator. In our example, this was dividing \(5x^3\) by \(x\), resulting in \(5x^2\).
• Next, multiply \(5x^2\) by \(x - 3\), giving \(5x^3 - 15x^2\).
• Subtract this from the original numerator to find the new polynomial to work with. Continue this process by dividing the new leading term by the leading term of the denominator, until all terms are accounted for.

This repetitive process works down the polynomial, creating a quotient \(Q(x) = 5x^2 + 17x + 48\) and a remainder \(144\). The final expression \(Q(x) + \frac{r(x)}{d(x)}\) helps in breaking down the original rational function more systematically.

The Remainder Theorem is a useful result when dividing one polynomial by another. It states that if you divide a polynomial \(P(x)\) by a polynomial \(d(x) = x - c\), the remainder of this division is \(P(c)\).

In this exercise, although we used long division to find the remainder, the Remainder Theorem provides us with an alternative method. After dividing by \(x - 3\), the remainder found was \(r(x) = 144\), confirming the consistency of our polynomial division where \(P(3)\) could provide similar insights.

Using the theorem, we evaluate the polynomial \(P(x) = 5x^3 + 2x^2 - 3x\) at \(x = 3\), offering another way to cross-verify results you get from long division. This interplay between the Remainder Theorem and polynomial long division underscores the fundamental concepts of algebra, ensuring that mathematical operations on polynomials remain accurate.