Problem 44 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{4 x^{2}+24 x}{x+6} $$

Step-by-Step Solution

Answer

The expression simplifies to \(4x\).

Step 1: Factor the Numerator

The first step is to factor the numerator, which is the expression \(4x^2 + 24x\). We notice that both terms have a common factor of \(4x\), so we factor \(4x\) out. This gives us: \[ 4x(x + 6). \]

Step 2: Identify Common Factors

Now, we have the expression \(\frac{4x(x+6)}{x+6}\). The next step is to identify any common factors in the numerator and the denominator.

Step 3: Simplify the Expression

Observe that \(x+6\) is a common factor in both the numerator and the denominator. Thus, we can cancel \(x+6\) from both parts of the fraction. After cancelling, the expression simplifies to \(4x\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step when working with algebraic expressions, especially when simplifying rational expressions. A polynomial might appear complex, but through factoring, it can be broken down into more manageable parts.

The first important point is to look for any terms that share a common factor, just like they have common ancestors. It’s like finding the greatest common factor (GCF) for numbers but within expressions with variables.

For the initial expression in our exercise, we have \(4x^2 + 24x\). Both terms include the factor of \(4x\). By taking \(4x\) out, we simplify the expression to \(4x(x + 6)\). This step is all about making expressions simpler and preparing them for further manipulation.

• Focus on each term within the polynomial.
• Pull out the largest common factor.
• Simplify the expression by writing it as a product of this factor and another expression.

Common Factors

After factoring a polynomial, the next step is to identify any common factors between the numerator and the denominator. This process is like looking for matching pairs of socks in a drawer.

In the expression \(\frac{4x(x+6)}{x+6}\), we conveniently find \((x+6)\) both above (numerator) and below (denominator) the fraction line. It's essential because these common factors can be canceled out, simplifying the expression further.

Here are some steps when looking for common factors:

• Carefully check each term in the numerator against those in the denominator.
• Look for entire expressions rather than just numbers or single variables.
• Mark the common elements so you can cancel them out easily.

Canceling Terms

Once we've identified the common factors in a rational expression, it's time to simplify it by canceling. Canceling is the process of removing the identical elements shared between the numerator and the denominator. This step is fundamental as it often leads to a much simpler form of the expression.

In our exercise, both the numerator and the denominator shared the term \((x + 6)\). By canceling \((x + 6)\), we are left with a simplified expression: \(4x\).

Canceling doesn't change the value of the expression, it only makes it easier to interpret and use.

Remember:

• Only cancel terms that appear identically in both the numerator and the denominator.
• Check your work by ensuring the original expression equals the simplified version when substituting values.
• Canceling helps reduce the risk of errors in calculations and provides clarity.