91Ó°ÊÓ

Polynomial expressions consist of terms which are algebraic expressions that can include variables, numbers, and exponents. In the exercise, you have a polynomial with three terms: \(5x^3y^3z^4\), \(25x^2y^3z^2\), and \(-35x^3y^2z^5\). Each of these terms is a product of the variables \(x\), \(y\), and \(z\), raised to different powers.

Polynomials can vary in complexity based on the number of terms and the powers of the variables involved. The expression in the exercise is a multi-variable polynomial with high degree terms, making it crucial to apply systematic methods to simplify or factor it. Understanding each term's structure helps in identifying common factors, which is our next topic.

The Greatest Common Factor (GCF) is the highest number or variable expression that divides each term of a polynomial without a remainder. Identifying the GCF is a key step in simplifying or factoring polynomial expressions. For the given polynomial, we look at each term:

• \(5x^3y^3z^4\)
• \(25x^2y^3z^2\)
• \(-35x^3y^2z^5\)

The coefficients are 5, 25, and -35. The GCF of these numbers is 5.

Next, consider the variables:

• The smallest power of \(x\) is \(x^2\).
• The smallest power of \(y\) is \(y^2\).
• The smallest power of \(z\) is \(z^2\).

Therefore, the GCF of the whole expression is \(5x^2y^2z^2\). This careful analysis allows us to extract the common factor, simplifying further calculations.

Factoring is the process of breaking down a complex expression into simpler components that, when multiplied together, give back the original expression. After identifying the GCF in our polynomial, the next step is to divide each term by \(5x^2y^2z^2\) to factor it out. This process simplifies the expression:

\[5x^2y^2z^2(xyz^2 + 5y - 7xz^3)\]

This technique—factoring by common factor—works well for expressions with recognizable common parts across terms. By confirming each term's divisibility by the extracted GCF, we ensure correctness. Expanding the factored form should yield the original expression, re-validating our steps. Proper use of factoring techniques greatly simplifies working with complex polynomial expressions, reducing potential for errors and opening paths for further algebraic manipulation or solving.