Question

Find the product. $$ \frac{x^2-3 x}{x-2} \cdot \frac{x^2+x-6}{x} $$

Short answer

The product of the given expressions \(\frac{x^2-3x}{x-2}\) and \(\frac{x^2+x-6}{x}\) is \(\frac{(x-3)(x+1)(x-6)}{(x-2)}\).

Step-by-step solution

Simplify

For the proper simplification, factorize the numerator of the first expression, \(x^2-3x\), to \(x(x-3)\), and the numerator of the second expression, \(x^2+x-6\), to \(x(x+1)(x-6)\). Thus, your rational expressions become: \(\frac{x(x-3)}{x-2} \cdot \frac{x(x+1)(x-6)}{x}\).

Multiply

To multiply the rational expressions, multiply the numerators together and the denominators together. This results to: \( \frac{x(x-3)(x+1)(x-6)}{x(x-2)}.\)

Reduce the Expression

There are two terms, x, one each in the numerator and the denominator that we can eliminate to reduce the expression. Therefore, the final simplified expression will be \( \frac{(x-3)(x+1)(x-6)}{(x-2)} \).

Key concepts

Simplify Rational Expressions

Understanding how to simplify rational expressions is a fundamental skill in algebra. A rational expression is a fraction where both the numerator and the denominator are polynomials. The objective is to simplify the expression into the least complex form possible. Simplifying can involve several steps, such as factoring polynomials, cancelling out common factors, and reducing to the lowest terms.

To simplify a rational expression, first factor both the numerator and the denominator to identify common factors. A common factor is a term that appears both in the numerator and the denominator. Simplification then involves dividing out the common factors, which essentially means 'cancelling' them. In the given exercise, after factoring, it was identified that the variable 'x' could be cancelled out since it appeared in both the numerator and the denominator.

Understanding this simplification process not only aids in finding a more manageable form of the expression but also prepares students for more complex problems involving rational expressions, such as those requiring addition or subtraction with common denominators.

Factorization in Algebra

The concept of factorization in algebra is about breaking down polynomials into the product of their simplest parts, known as factors. It's similar to finding the prime factorization of a number, where a number is expressed as a product of its prime factors. In algebra, factorization is used with polynomial expressions.

Tips for Factorization:

• Identify a common factor: Check if all terms in the polynomial have a common monomial component.
• Use known patterns: Recognize and apply special product formulas like the difference of squares, perfect square trinomials, or sum and difference of cubes.
• Apply techniques: Use methods such as grouping or the polynomial gcd (greatest common divisor) to simplify.

Once a polynomial is factored, it becomes much easier to perform operations such as multiplication and division with other polynomials, as seen in the exercise. Correct factorization is crucial as it leads to the exact cancellation of terms and simplification of the expression.

Polynomial Division

Dividing polynomials, also known as polynomial division, can be carried out using various methods depending on the complexity of the polynomials involved. The two primary methods are long division and synthetic division, both aiming to simplify polynomials or solve for zeros.

When it comes to rational expressions, polynomial division is often used when the polynomial in the numerator is divided by the polynomial in the denominator, just like numerical division. In our example, after multiplication, we effectively carry out a form of polynomial division when we cancel out the common 'x'. We avoid more tedious long division since we have already factored and hence, reduced the expression. Thus, understanding how to divide polynomials is integral to working with rational expressions and can significantly simplify more complex algebraic operations.