Problem 75 - Algebra 2

Question

Simplify each expression. What are the restrictions on the variable? $$ \frac{3 x}{6 x^{2}-9 x^{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{2 - 3 x^{3}}\) and the restriction on the variable is \(x \neq \sqrt[3]{\frac{2}{3}}\).

Step 1: Factorise the denominator

Start by factoring out the common factor of \(x\) in the denominator. The expression becomes \(\frac{3 x}{x(6 x - 9 x^{4})}\).

Step 2: Cancel out common factors

Next, cancel out the common factor of \(x\) from the numerator and the denominator. After doing this, the resulting expression becomes \(\frac{3}{6 - 9 x^{3}}\).

Step 3: Simplify the expression

Further simplify the expression by recognising that 3 can cancel with the 6 in the denominator: \(\frac{1}{2 - 3 x^{3}}\). This is the simplest form.

Step 4: Find the restrictions on the variable

The denominator cannot be zero, so \(2 - 3 x^{3}\) cannot be 0. Solve the equation \(2 - 3 x^{3} = 0\) for \(x\) to find the restrictions on the variable. This gives \(x = \sqrt[3]{\frac{2}{3}}\) as the restriction on the variable.

Key Concepts

Factoring Polynomials

One of the essential skills in algebra is factoring polynomials. This process involves breaking down a polynomial into simpler components, usually products of smaller polynomials or numbers. It is similar to finding the prime factors of a number. In the expression \( \frac{3x}{6x^2 - 9x^5} \), we notice that the terms in the denominator share common factors. By factoring out the greatest common factor, we can start simplifying. In this case, the greatest common factor is \(x\). Factoring \(x\) out of the denominator, we can rewrite it as \(x(6x - 9x^4)\). This step is crucial because it sets the stage for further simplification. Always look for shared factors, and taking them out is often your first move in simplifying expressions.

Simplifying Algebraic Expressions

Once a polynomial has been factored, the next step is simplifying the expression. This means reducing it to its simplest form. In algebraic expressions like \( \frac{3x}{6x^2 - 9x^5} \), once you have factored the denominator as \(x(6x - 9x^4)\), you can cancel out common factors from the numerator and denominator. Here, the common factor \(x\) can be canceled out, leaving us with the expression \( \frac{3}{6 - 9x^3} \).

• Always check for and cancel common factors to simplify.
• After canceling, simplify numerical coefficients by dividing through by their greatest common divisor.

The 3 in the new numerator and the 6 in the denominator can further be reduced to \( \frac{1}{2 - 3x^3} \). This is the simplest form of the expression.

Variable Restrictions

In algebra, it's essential to identify any restrictions on the variable within a rational expression. Restrictions occur where the denominator becomes zero, making the expression undefined. For the simplified expression \( \frac{1}{2 - 3x^3} \), we need to ensure the denominator doesn't equal zero. To find restrictions, solve the equation \(2 - 3x^3 = 0\) for \(x\). By setting the denominator equal to zero, we find that when \(x = \sqrt[3]{\frac{2}{3}}\), the denominator becomes zero.

• Ensuring the denominator is not zero is crucial in rational expressions.
• Solve for the variable to find where these restrictions apply.

If the solution to your denominator equation is a real number, it translates to a restriction on what values \(x\) can take.