Problem 23 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{26 y^{3}+13 y^{2}+39 y}{13 y} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(2y^2 + y + 3\).

Step 1: Divide each term of the numerator by the denominator

We have the following rational function: $$ \frac{26 y^{3}+13 y^{2}+39 y}{13 y} $$ To perform the division, divide each term of the numerator by the denominator, which is \(13y\). So: $$ \frac{26y^3}{13y} + \frac{13y^2}{13y} + \frac{39y}{13y} $$

Step 2: Simplify each term

Now, we will simplify each term: - \(\frac{26y^3}{13y} = 2y^2\) (divide coefficients and subtract exponent: \(3-1=2\)) - \(\frac{13y^2}{13y} = y\) (divide the coefficient, and subtract exponent: \(2-1=1\)) - \(\frac{39y}{13y} = 3\) (divide coefficients, the variable \(y\) cancels out)

Step 3: Combine the simplified terms

Now, let's write the result of the division by combining the simplified terms: $$ 2y^2 + y + 3 $$ So, the division of the given rational function simplifies to: $$ \frac{26 y^{3}+13 y^{2}+39 y}{13 y} = 2y^2 + y + 3 $$

Key Concepts

Rational Functions

Rational functions are mathematical expressions that represent the division of two polynomials. They take the form \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomial functions. A key characteristic of rational functions is their potential to be simplified by dividing the numerator and the denominator by common factors.

In our exercise, we work with the rational function \( \frac{26 y^{3}+13 y^{2}+39 y}{13 y} \). The goal is to simplify this expression to its simplest form by dividing each term of the numerator by the denominator, a process which allows us to remove common factors and unveil a more straightforward representation of the function. Simplifying rational functions is a fundamental skill in algebra as it helps clarify the behavior of the function, such as identifying asymptotes and intercepts.

Simplifying Expressions

When it comes to simplifying mathematical expressions, the objective is to condense them into their most elemental form while maintaining their original value. Simplification is not only about aesthetics; it makes expressions easier to work with and understand.

For example, in simplifying \( \frac{26y^3}{13y} \), we focus on reducing the expression to its lowest terms by dividing the coefficients \( (26 ÷ 13 = 2) \) and by employing the laws of exponents to subtract the powers \( (3 - 1 = 2) \) when dividing like bases. The result is a concise and efficient representation of the original expression. Approaching each term individually and then combining them yields the most simplified form of the expression.

Algebraic Division

Algebraic division, much like arithmetic division, requires dividing terms to break them down into simpler components. It is particularly useful in dealing with polynomials and rational functions. In the provided exercise, we divided a polynomial by a monomial term-by-term, which is a common algebraic division technique.

To do so effectively, there are steps to follow: identify each term in the numerator, divide each by the denominator, simplify, and then recombine the results. Mastery of algebraic division also facilitates the understanding of more complex concepts, such as polynomial long division, synthetic division, and the division of polynomials with multiple variables.

Understanding algebraic division is beneficial not just in simplifying expressions but also in solving equations, factoring polynomials, and analyzing function behavior.