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When factoring polynomials, the Greatest Common Factor (GCF) is the largest factor that divides all terms of the polynomial. To find the GCF for both coefficients and variables, perform these steps:

For the coefficients, list the factors of each number. For example, the factors of 120 include 1, 2, 3, 4, 5, 6, etc., and the factors of 48 include 1, 2, 3, 4, 6, etc. The highest number shared between the two lists will be the GCF. For 120 and 48, that number is 24.

For the variables, look at the lowest power (exponent). In the polynomial 120y^6 + 48y^4, the term with the smallest power of y is y^4. Hence, the GCF for the variables is y^4. Once determined, the overall GCF for the polynomial is found by combining the GCF of coefficients and variables. For this example, it is 24y^4.

Polynomial coefficients are the numerical parts of each term in the polynomial. They play a crucial role in identifying the GCF. In our example, the coefficients are 120 and 48.
To factor polynomials, start by identifying these coefficients and finding their GCF. List the multiples of each number, and find the highest common number.

• For 120: the factors are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.


• For 48: the factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

Since both lists have 24 as the highest common factor, it is the GCF. Once the GCF of the coefficients is found, the coefficients in the polynomial can be simplified by dividing them by this GCF. This is a fundamental step in the factoring process.

Exponents determine how many times a variable is multiplied by itself. They are crucial in factoring polynomials, as the GCF for the variables depends on the lowest exponent present. In the polynomial 120y^6 + 48y^4, we have exponents 6 and 4.

• First, identify these exponents: 6 and 4.


• Then, find the smallest one. Here, 4 is smaller than 6.

Hence, the GCF for the variable part of the polynomial is y^4. Multiplying by this GCF allows you to simplify and factor out from the polynomial. Combining this with the GCF of the coefficients gives you a common factor for the entire polynomial: 24y^4.

By dividing each term by this GCF, you simplify the process and can easily factor the polynomial, as shown in the solution, resulting in 24y^4(5y^2 + 2).