Problem 27 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$ \frac{42 x-6 x^{3}}{36 x} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{(7 - x^{2})}{6}\)

Step 1: Factor out common factors

Start by factoring out the common factors on the numerator and the denominator independently. In the numerator, you can factor out \(6x\) to get \(6x(7 - x^{2})\). The denominator already is \(36x\). This simplifies to \(\frac{6x(7 - x^{2})}{36x}\).

Step 2: Canceling common factors

To simplify the expression further, cancel out the common terms from the numerator and the denominator. In this case, you can simplify \(6x\) from the numerator and the denominator, and divide 36 by 6 to get 6. This will leave you with \(\frac{(7 - x^{2})}{6}\).

Key Concepts

Factoring

Factoring is a crucial technique in algebra, especially when working with rational expressions. It involves breaking down expressions into smaller components, known as factors, that are multiplied together to form the original expression.For instance, in the expression \(42x - 6x^3\), the term \(6x\) can be factored out, bringing it outside the parentheses. Thus, you get \(6x(7 - x^2)\).

• This idea rests on finding the greatest common factor (GCF), which is the largest factor shared by the terms.
• In our example, both 42x and 6x^3 share 6x, hence it's factored out.

This process simplifies the expression and is often the first step in tackling problems involving rational expressions.Being able to spot factors quickly can make solving math problems much easier.

Canceling Common Factors

Canceling common factors is like cleaning up an expression to make it simpler and more concise.Once you factor out the expressions, as seen with our example \(\frac{6x(7 - x^2)}{36x}\), you can look for terms that are common both in the numerator and the denominator. In this specific case, the common factor is \(6x\). By canceling it from both the numerator and the denominator, you simplify the expression even further.

• Reducing these common factors to one ensures you end up with the simplest form of the expression.
• Here, the expression reduces to \(\frac{(7 - x^2)}{6}\) after canceling, highlighting how crucial the canceling step is.

This step is vital because it ensures the expression is as reduced as possible, minimizing errors or misinterpretations in future computations.

Rational Expressions

Rational expressions are fractions that contain polynomials in either the numerator, the denominator, or both.Handling rational expressions can sometimes seem challenging, but the process becomes manageable when broken down into basic steps of factoring and simplifying.

• Before simplifying, always ensure you factor out any common terms.
• Next, cancel out these common factors to achieve the simplest form of the expression.
• In the example \(\frac{42x-6x^{3}}{36x}\), the rational expression is simplified through factoring and canceling common terms.

Understanding rational expressions helps in solving various mathematical problems with efficiency. This process also integrates into more complex equations, hence mastering it opens doors to tackling more advanced algebraic challenges.