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The greatest common factor (GCF) is the largest factor that can divide all terms in an expression without leaving a remainder. To find the GCF of an expression, let's look at the terms and identify the common factors. For example, consider the terms 98r³ and 72r. We first determine the GCF of the numerical coefficients (98 and 72). We find that the GCF of 98 and 72 is 2. Next, we look at the variable part. The smallest power of the variable r in both terms is r (or r¹). Therefore, the GCF of 98r³ and 72r is 2r.Once you find the GCF, you can factor it out of the expression. In this case, we factor out 2r from 98r³ - 72r, resulting in:\[98r³ - 72r = 2r(49r² - 36)\]

The difference of squares is a specific type of polynomial that can be factored into two binomials. The general form of a difference of squares is:\[a² - b² = (a + b)(a - b)\]In this form, 'a' and 'b' are both squared terms. Let's apply this to our problem. After factoring out the GCF, we're left with 49r² - 36. Notice that 49r² and 36 are both perfect squares. We can write this as:\[(7r)² - 6²\]Here, 'a' is 7r and 'b' is 6. According to the difference of squares formula, we can factor this further:\[49r² - 36 = (7r + 6)(7r - 6)\]

Factoring polynomials involves breaking down a complex polynomial into simpler expressions that can be multiplied to obtain the original polynomial. This simplifies solving and analyzing the polynomial. Let's walk through the process with our given expression:1. **Identify and factor out the GCF**: First, find the greatest common factor (GCF) of the terms. For 98r³ - 72r, the GCF is 2r. Factor out the 2r:\[98r³ - 72r = 2r(49r² - 36)\]2. **Factor the remaining polynomial**: Next, look at the polynomial 49r² - 36 left inside the parentheses. Notice that it is a difference of squares. The difference of squares formula is \(a² - b² = (a + b)(a - b)\). Here, 49r² is (7r)² and 36 is 6². Use the difference of squares method to factor further:\[49r² - 36 = (7r + 6)(7r - 6)\]Putting it all together, the fully factored form of the original polynomial is:\[98r³ - 72r = 2r(7r + 6)(7r - 6)\]By understanding the GCF, recognizing patterns like the difference of squares, and systematically breaking down polynomials, you can easily factor them and simplify complex algebraic expressions.