91Ó°ÊÓ

To begin factoring polynomials, the first step is to determine the Greatest Common Factor (GCF). The GCF is the largest factor that is common to all terms in the polynomial. This includes considering both the coefficients (numerical parts) and the variables with their exponents.
To find the GCF, follow these small steps:

• List the factors of each coefficient.
• Determine the smallest power of each variable present in all terms.
• Identify the common factors.

For example, in the polynomial expression:
\( 20a^{3}b^{6}, -3a^{2}b^{4}, -2ab^{2} \)
the GCF of the coefficients 20, -3, and -2 is simply 1, because they don't share any common factor larger than 1. For the variables, the smallest power of a is \(a\), and the smallest power of b is \(b^{2}\). Therefore, the overall GCF of the polynomial is \(a b^{2}\).

A polynomial expression is a sum of terms, each consisting of a coefficient multiplied by variables raised to non-negative integer exponents. In general, a polynomial in one variable, \(x\), might look like:
\(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\)

Here, each \(a_i\) is a coefficient, and the powers of \(x\) decrease from left to right. In the problem:
\( 20a^{3}b^{6} - 3a^{2}b^{4} - 2ab^{2}\)

we have a polynomial in two variables, \(a\) and \(b\). Notice that each term is a product of a coefficient (like 20, -3, or -2) and a combination of \(a\) and \(b\) raised to different powers.
Understanding the structure of polynomial expressions is crucial for factoring and simplifying them effectively.

After identifying the GCF in a polynomial, the next step is variable factorization. This involves 'pulling out' the GCF from the polynomial, simplifying the expression.
Let's do this with our example:
\( 20a^{3}b^{6} - 3a^{2}b^{4} - 2ab^{2} \).
We determined that the GCF is \(ab^{2}\).

To factor \(ab^{2}\) out of each term, we write:
\[ 20a^{3}b^{6} ÷ (ab^{2}) = 20a^{2}b^{4}, -3a^{2}b^{4} ÷ (ab^{2}) = -3ab^{2}, -2ab^{2} ÷ (ab^{2}) = -2.\]

This yields:
\(20a^{3}b^{6} - 3a^{2}b^{4} - 2ab^{2} = ab^{2}(20a^{2}b^{4} - 3ab^{2} - 2)\).

This is the factored form of the polynomial. When factoring polynomials, you always want to check if the remaining expression in the parentheses can be factored further. In this case, it is already in its simplest form.