91Ó°ÊÓ

Before you start factoring an algebraic expression, it’s essential to identify the common factors in your terms. This is the first and often crucial step in the factoring process. A common factor is a number or variable that divides exactly into each term of the expression. In the given problem, \(7x^2 - 21x\), both terms share the common factors of 7 and x. Finding this common factor is similar to finding the greatest common divisor (GCD) in basic arithmetic.

**Why is it important to find the common factor?**

• It simplifies the expression, making it easier to work with.
• It helps in breaking down complex expressions into more manageable parts.
• It prepares the expression for more advanced factoring techniques, if necessary.

Once you’ve identified the common factor, the next step is to factor it out of the expression. In our example, the common factor is 7x. When we factor this out, each term in the expression is divided by 7x.

Here’s how we do it step-by-step:
1. **Start with the original expression:** \(7x^2 - 21x\).
2. **Identify the common factor:** In this case, it’s 7x.
3. **Factor out (or divide) 7x from each term:** \[7x^2 - 21x = 7x(x) - 7x(3)\]
4. **Combine the factors:** The factored form is \[7x(x - 3)\]

By following these steps, you’ll simplify expressions correctly and efficiently. This method is especially useful for more complicated expressions, ensuring you don't miss any common factors.

After factoring the expression, it's crucial to verify that your factored form is correct. Verifying ensures that you haven't missed any steps or made errors.

**To verify:** Distribute the common factor back into the factored form and check if it matches the original expression.

For our example, take the factored form \[7x(x - 3)\]. Let's distribute 7x back:

• \(7x \times x = 7x^2\)
• \(7x \times (-3) = -21x\)

Combining these terms, we get \[7x^2 - 21x\], which matches the original expression. This confirms that our factored form is correct.

Verification is a good habit because:

• It helps catch any mistakes early.
• It builds confidence in your algebraic manipulation skills.
• It solidifies your understanding of the process.