91Ó°ÊÓ

When working with polynomials, finding the Greatest Common Factor (GCF) is the first step in simplification. The GCF is the highest number or greatest polynomial term that divides into each term of the given expression without leaving a remainder. In the exercise you provided, the terms in the numerator are \(9a^3b^2\) and \(-18a^2b^3\). To find their GCF, first look at the coefficients 9 and 18. The greatest common factor of these numbers is 9. Then, examine the variables:- For \(a^3\) and \(a^2\), the GCF is \(a^2\) because it is the smallest power of \(a\) appearing as a factor in both terms.- For \(b^2\) and \(b^3\), the GCF is \(b^2\) because it is the smallest power of \(b\) that divides both terms.Thus, the GCF for the entire expression is \(9a^2b^2\). By factoring this out, you create a simpler expression that is easier to work with.

Factoring polynomials involves expressing a polynomial as a product of its factors. It simplifies the polynomial and makes it easier to manipulate or simplify further, as in the case of our given exercise.Once the GCF \(9a^2b^2\) was factored out from the numerator \(9a^3b^2 - 18a^2b^3\), it allowed us to rewrite the expression in a simplified form: \[9a^2b^2(a - 2b)\] This factored expression reveals any potential for simplification in subsequent operations involving algebraic expressions.- The term \(a - 2b\) is known as the factored form of the polynomial \(a^3b^2 - 2a^2b^3\) since it simplifies the original polynomial and often helps in cancelling terms elsewhere, as we see in rational expressions.

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying these expressions can often appear complex, but understanding the basics of factoring and finding common factors can significantly ease the process.In the exercise given, the rational expression is:\[ \frac{9a^3b^2 - 18a^2b^3}{3a^2b} \]By factoring the numerator, we simplify it into a form \(9a^2b^2(a - 2b)\). This allows for cancellation of matching terms in the numerator and denominator. Specifically, the \(a^2\) and \(b\) from the denominator cancel with parts of \(9a^2b^2\) in the numerator. This reduction simplifies the fraction to a simpler expression:\[ 3b(a - 2b) \] Ultimately, simplifying rational expressions involves:

• Factoring the numerator and the denominator to their simplest forms.
• Canceling any common factors present in both the numerator and the denominator.

This reduces the complexity of the expression and results in a cleaner, more manageable form.