Problem 39 - Prealgebra and Introductory Algebra

Question

Multiply. $$\frac{3 x^{2}+2 x}{2 x y-3 y} \cdot \frac{2 x y^{3}-3 y^{3}}{3 x^{3}+2 x^{2}}$$

Step-by-Step Solution

Answer

The final answer after multiplying the given two rational expressions is \( \frac{-5x^{2}y^{3}+6x^{3}y^{3}}{-5x^{3}y-2x^{2}y} \)

Step 1: Factorize each expression

Start by checking if it's possible to factorize each expression. \nThe terms in each expression are already simplified so we can move to the next step.

Step 2: Multiply the rational expressions

Then multiply the numerators together and the denominators together. \n So, \( (3x^{2}+2x) \times (2xy^{3}-3y^{3}) \) becomes \(6x^{3}y^{3}-9x^{2}y^{3}+4x^{2}y^{3}-6xy^{3}\) in the Numerator. \n And, \( (2xy-3y) \times (3x^{3}+2x^{2}) \) becomes \(6x^{4}y+4x^{3}y-9x^{3}y-6x^{2}y\) in the Denominator.

Step 3: Simplify the expression

Rearrange and combine like terms in numerator and denominator. \n So, the final expression becomes \( \frac{-5x^{2}y^{3}+6x^{3}y^{3}}{-5x^{3}y-2x^{2}y} \)

Key Concepts

Rational Expressions

Rational expressions might sound complex, but they really just involve fractions where both the numerator and the denominator are polynomials. In our exercise, for example, we have terms like \( \frac{3x^{2} + 2x}{2xy - 3y} \). The polynomial at the top of the fraction is the numerator, and the one at the bottom is the denominator.

• The core idea behind rational expressions is quite similar to regular fractions: just as numbers can be expressed in terms like \( \frac{1}{2} \) or \( \frac{3}{4} \), polynomials can also be written in the same form.
• Working with rational expressions involves many of the same principles as working with numerical fractions, such as simplifying and factoring.

This understanding sets the stage for more operations, just like you would add, subtract, multiply, or divide numerical fractions.

Factorization

Factorization is a critical process when dealing with polynomials, and it helps simplify rational expressions efficiently. Factorization involves breaking down a complex expression into parts (or factors) that when multiplied together give back the original expression.

In the exercise provided, factorization checks if the given polynomials can be simplified or rewritten as simpler multiplicative parts. Although our example did not require further factorization because it was already simplified, this is a crucial preliminary step.

• Factorization often involves finding common terms or factors within a polynomial.
• For instance, if you have \( x^{2} - x \), you can factor it out to \( x(x - 1) \), breaking it down using a common factor.

The purpose of factorizing before performing any operations is to ensure that any possible simplification is apparent, making subsequent calculations more straightforward.

Simplification

Simplification plays a pivotal role in making rational expressions more digestible. Once multiplication is completed, the next step is often to simplify by combining like terms and reducing if possible.

In the exercise, after multiplying the numerators and denominators, the expression becomes quite congested with terms such as \(6x^{3}y^{3} - 9x^{2}y^{3} + 4x^{2}y^{3} - 6xy^{3}\). Simplifying involves:

• Combining like terms, which are terms that contain the same variables to the same power (e.g., \(x^{2}y^{3}\) terms together).
• Reducting unnecessary complexity in the rational expression.

The goal is always to reach the simplest form possible, where no further simplification can be achieved, which makes both interpretation and further operations much easier.

Numerators and Denominators

When handling rational expressions, it's crucial to have a firm grasp of numerators and denominators, as these are the backbone of any fraction.

In polynomial multiplication, just as in the exercise shown, we multiply the numerators together to form a new numerator and the denominators together to form a new denominator. Here is how it works:

• The numerator of a rational expression is the top part of the fraction, representing one or more polynomials; for instance, \( 3x^{2} + 2x \) in our problem set.
• The denominator is the bottom part and also a polynomial, like \( 2xy - 3y \).

Understanding how these interact is vital because simplifying often involves canceling out common factors between numerators and denominators, leading to significantly simpler expressions.