Problem 15 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{4 x^{3}}{2 x} $$

Step-by-Step Solution

Answer

Question: Simplify the following rational expression: $$ \frac{4x^3}{2x} $$ Answer: The simplified rational expression is: $$ 2x^2 $$

Step 1: Identify common factors

We can see that both the numerator (4x^3) and the denominator (2x) have a common factor of x. Additionally, 4 and 2 are both divisible by 2 which is also a common factor.

Step 2: Factor out the common factors

Divide the numerator and the denominator by the common factors (2x). $$ \frac{4 x^{3}}{2 x} = \frac{2 \cdot 2x^{3}}{2 \cdot x} $$

Step 3: Cancel common factors

Cancel out the common factors in the numerator and denominator. $$ \frac{2 \cdot 2x^{3}}{2 \cdot x} = \cancel{2} \frac{2x^{3}}{\cancel{2}x} = \frac{2x^3}{x} $$

Step 4: Simplify the expression

Simplify the expression by dividing the exponent of x in the numerator by the exponent of x in the denominator and applying the power rule for exponents. $$ \frac{2x^3}{x} = 2x^{3-1} = 2x^2 $$ The simplified rational expression in lowest terms is: $$ 2x^2 $$

Key Concepts

Factoring

When simplifying rational expressions, factoring is a crucial step. Think of factoring as breaking down numbers or expressions into simpler units that can be multiplied together to obtain the original number or expression. For instance, to factor the numerator in the equation \( \frac{4x^3}{2x} \), we identify that 4 can be written as \( 2 \times 2 \) and \( x^3 \) as \( x \times x \times x \). The denominator 2x can be simplified to \( 2 \times x \). This way, the expression \( \frac{4x^3}{2x} \) can also be seen as \( \frac{2 \cdot 2 \cdot x^3}{2 \cdot x} \). By factoring both the numerator and the denominator, it makes it easier to identify common factors that can be cancelled.

Common Factors

Common factors are elements that appear in both the numerator and the denominator of a rational expression. Recognizing these factors is key to simplifying the expression. In our example, both the numerator \( 4x^3 \) and denominator \( 2x \) have the common factor \( 2x \). This means both terms can be divided by \( 2x \), simplifying the expression significantly.

• For numeric coefficients, find numbers that divide both evenly, like 2 in our case, for 4 and 2.
• For variable parts, identify the lowest power of the variable that appears in both the numerator and denominator, such as \( x \).

This identification allows us to cancel terms and streamline the expression.

Canceling Terms

Canceling terms is the process where we remove the common factors from both the numerator and denominator. This simplifies the rational expression without changing its value. Let's consider our expression \( \frac{2 \cdot 2x^3}{2 \cdot x} \). Here, we cancel the \( 2 \) and \( x \) that appear in both. So, \( \cancel{2} \cdot 2x^3 \) over \( \cancel{2} \cdot x \) becomes \( 2x^3 \) over \( x \). When canceling, it's vital to ensure both numerator and denominator contain the factor to be canceled to maintain equivalent expressions.

Power Rule for Exponents

The power rule for exponents is a powerful tool when simplifying terms within rational expressions. This rule tells us that when you divide like bases with exponents, you subtract the exponent of the denominator from that of the numerator. For the expression \( \frac{2x^3}{x} \), we apply this rule:

• Notice the base \( x \) and exponents 3 in the numerator and 1 in the denominator.
• Subtract the exponents: \( 3 - 1 = 2 \).

So, \( \frac{2 \cdot x^3}{x} \) simplifies to \( 2x^{3-1} \) or \( 2x^2 \). Remember, this only applies when bases are the same, making it a handy trick in simplifying terms.