Problem 32 - Elementary Algebra

Question

Factor out the GCF. $$ 5 x-10 $$

Step-by-Step Solution

Answer

The factored form is \(5(x - 2)\).

Step 1: Identify the GCF

The given expression is \(5x - 10\). First, we identify the Greatest Common Factor (GCF) of the terms. The coefficients of the terms are 5 and 10. The GCF of 5 and 10 is 5. In terms of variables, both terms have no common variables as \(10\) can be considered as \(10x^0\). Therefore, the GCF of the expression is 5.

Step 2: Factor Out the GCF

We now factor out the GCF, which is 5, from both terms in the expression \(5x - 10\). When we divide each term by 5, we get: \[ 5x \div 5 = x \] \[ -10 \div 5 = -2 \]Thus, factoring out gives us: \[ 5(x - 2) \]

Step 3: Verify the Factored Expression

To ensure our factoring is correct, we distribute back what we factored out: \[ 5(x - 2) = 5x - 10 \]This results in the original expression, confirming that \(5(x - 2)\) is indeed the correctly factored form of \(5x - 10\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number that evenly divides two or more numbers. It's essential when factoring expressions as it simplifies the process. In the expression \(5x - 10\), we look at the numbers 5 and 10. Both can be divided by 5, which is their GCF. By factoring it out, we reduce the expression to a simpler form.
Understanding the GCF helps in breaking down complex expressions more manageably:

• Check each term's numerical coefficient.
• Identify the highest number that divides them all.
• If there are variables involved, check for the common variable factor as well.

In our case, there are no common variables between the terms since "x" is absent in the second term as \(10x^0\). So, 5 is the GCF for the numerical part alone.

Factoring Expressions

Factoring is about pulling out the GCF from each term to simplify the expression. This step transforms the expression into a product of factors, which can be easier to work with or solve.
For \(5x - 10\), we identified that 5 is the GCF. By dividing each term in the expression by 5, we factor out the GCF:

• \(5x \div 5 = x\)
• \(-10 \div 5 = -2\)

Thus, the factored expression becomes \(5(x - 2)\). Factoring expressions is often used to simplify equations, make them easier to solve, or for further algebraic operations.

Verifying Factored Expressions

Verification ensures we've factored an expression correctly, preserving its value. To verify, distribute the factor back through the expression. In this case, multiply 5 back through \(x - 2\):

• \(5 \cdot x = 5x\)
• \(5 \cdot (-2) = -10\)

Combine these results to confirm they match the original expression \(5x - 10\).
Verification is your safety net that provides confidence your work is correct. It’s particularly useful when the steps become more complex. With practice, this becomes a quick check in your mind’s calculations.