Problem 65 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{20 x^{3}-20 x^{4}}{x^{2}-2 x+1} $$

Step-by-Step Solution

Answer

The simplified form is \(-\frac{20x^3}{x - 1}\).

Step 1: Identify the common factor in the numerator

Look at the terms in the numerator, \(20x^3\) and \(-20x^4\). Both terms have a common factor of \(20x^3\). Factor \(20x^3\) out of the numerator: \(20x^3(1 - x)\).

Step 2: Factor the denominator

The denominator is \(x^2 - 2x + 1\), which is a perfect square trinomial. It can be factored as \((x - 1)^2\).

Step 3: Write the simplified expression

Replace the factored terms into the rational expression: \(\frac{20x^3(1 - x)}{(x - 1)^2}\).

Step 4: Simplify the expression further

Notice that \((1 - x)\) is the same as \(-(x - 1)\), allowing us to write the expression as \(\frac{-20x^3(x - 1)}{(x - 1)^2}\). This lets us cancel one \((x - 1)\) from the numerator and denominator: \(\frac{-20x^3}{x - 1}\).

Step 5: Final simplified form

The expression is simplified to \(-\frac{20x^3}{x - 1}\).

Key Concepts

Factoring Polynomials

When we talk about factoring polynomials, it means breaking down a complex expression into simpler factors that, when multiplied together, give back the original polynomial. It’s similar to finding the prime factors of a number but in a more algebraic way.
To factor the expression, first, identify if there are any common elements in each term. In our example, the numerator is made up of two terms: \(20x^3\) and \(-20x^4\). Notice that both terms include \(20x^3\). By factoring out this common term, the expression becomes much simpler: \(20x^3(1 - x)\).
This technique helps to break down an expression into its basic building blocks. It not only simplifies the polynomial but makes the entire rational expression easier to manage.

Perfect Square Trinomial

Understanding what a perfect square trinomial is can make factoring much easier. A perfect square trinomial is a specific type of trinomial that can be factored into a binomial squared.
The standard form of a perfect square trinomial is \(a^2 - 2ab + b^2\). It can always be rewritten as \((a - b)^2\). This special structure is what allows us to factor the denominator of our original expression, \(x^2 - 2x + 1\), into \((x - 1)^2\).
Recognizing these patterns makes factoring quick and efficient, leading to more manageable expressions. This knowledge can be applied to many different algebra problems, making it an essential tool in simplifying rational expressions.

Common Factor

A common factor is a term that divides exactly into each of the terms in a polynomial or a larger expression. It is a crucial concept in simplifying expressions because it allows us to reduce overall complexity by cancelling it out.
In our example, the common factor in the numerator \(20x^3\) appears in both terms \(20x^3\) and \(-20x^4\). By identifying and extracting this factor, the expression transform into a simpler one: \(20x^3(1 - x)\).

• This makes the expression easier to handle.
• It also highlights the repetitive structure and simplifies further cancellations with the denominator.

Therefore, finding the common factor is often the first step in simplifying any rational expression or polynomial.

Canceling Terms

Canceling terms is the process of reducing an expression by eliminating terms that appear both in the numerator and the denominator. This technique significantly simplifies the rational expression.
In the given problem, the expression after factoring becomes \(\frac{-20x^3(x - 1)}{(x - 1)^2}\). Here, \((x - 1)\) is common in both the numerator and the denominator. By canceling one \((x - 1)\) from both parts, the expression simplifies to \(\frac{-20x^3}{x - 1}\).
This cancellation helps to streamline the expression into its simplest form. However, it’s important to ensure that the expression being canceled out is not equal to zero in its domain, as this could lead to undefined expressions.