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A common factor is a number or variable that divides exactly into two or more other numbers or variables. In the expression \(7b^{2}h - 28b\), identifying common factors is the first step in the process of factoring polynomials. Here, both terms in the expression have coefficients and variables. To find a common factor, you need to look for values that can be multiplied to obtain each term in the expression.

• For the term \(7b^{2}h\), the factors are \(7\), \(b\), and \(h\).
• For the term \(-28b\), the factors are \(-28\) and \(b\).

Both terms share the common factor of \(7b\), which is a crucial step in rewriting or transforming the expression. Recognizing common factors helps in simplifying expressions and is a foundational skill for more complex factoring techniques.

Finding the greatest common factor (GCF) involves determining the highest factor that both terms share. In our expression \(7b^{2}h - 28b\), you first identify the common factors of each term, which we found to be \(7b\). The GCF is essentially the largest of these shared factors.

• The GCF includes the greatest numerical factor. Between \(7\) and \(28\), it's \(7\).
• The GCF must also include all shared variables with the lowest powers. Here, both terms include at least one \(b\).

By extracting the GCF: \(7b\), you rewrite the expression in a factored form: \(7b(bh - 4)\). This process of finding and using the GCF helps streamline polynomial expressions, making them simpler to work with or solve in equations.

Simplifying expressions involves rewriting them in a more condensed and efficient manner while maintaining the same value or equivalence. After factoring out the greatest common factor (GCF) in our polynomial, we simplify the expression \(7b^{2}h - 28b\) into \(7b(bh - 4)\). This process involves a few confirmation steps:- **Distribute to verify**: Multiply the factored expression to ensure it returns to its original form to avoid errors. Checking \(7b(bh - 4)\) by distribution gives us \(7b^{2}h - 28b\), confirming the simplification is accurate.- **Efficiency**: The simplified version often makes it easier to handle when inserted into further equations, compared to handling more complex initial terms.Recognizing how simplifying expressions works can aid students in solving more challenging algebraic problems with greater ease. Factoring to simplify reveals how expressions can be broken down and structured for accuracy and convenience.