Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-8,\) write each polynomial as the product of its greatest common monomial factor and a polynomial. $$ x^{3} y^{3}-2 x^{3} y^{2}+x^{2} y^{2} $$

Step-by-Step Solution

Answer

The polynomial can be written as \(x^2 y^2 (xy - 2x + 1)\).

Step 1: Identify the Terms

Examine the given polynomial: \(x^{3} y^{3} - 2x^{3} y^{2} + x^{2} y^{2}\), which consists of three terms.

Step 2: Determine the Greatest Common Factor (GCF)

For the polynomial \(x^{3} y^{3} - 2x^{3} y^{2} + x^{2} y^{2}\), determine the greatest common monomial factor by finding the lowest degree of each variable in all terms. The GCF is \(x^2 y^2\).

Step 3: Factor Out the GCF

Divide each term of the polynomial by the GCF \(x^2 y^2\):\[ \frac{x^{3}y^{3}}{x^{2}y^{2}} = xy, \frac{-2x^{3}y^{2}}{x^{2}y^{2}} = -2x, \frac{x^{2}y^{2}}{x^{2}y^{2}} = 1\]Thus, the polynomial can be expressed as \(x^2 y^2 (xy - 2x + 1)\).

Step 4: Reconstruct the Expression

Write the original polynomial as the product of the GCF and the resulting polynomial from Step 3: \(x^2 y^2 (xy - 2x + 1)\).

Key Concepts

Greatest Common Factor

When factoring polynomials, one of the key first steps is finding the Greatest Common Factor (GCF). The GCF of a set of terms is the largest expression that can evenly divide each term. To determine the GCF for our polynomial, we focus on the coefficients and variables in each term.

• Check the numerical coefficients: Here, the terms have coefficients of 1, -2, and 1. The GCF for these numbers is 1 because it is the only integer that divides each without leaving a remainder.
• Find the lowest power of each variable present in every term: For this polynomial, we compare the powers of each variable.
  • For variable x, the minimum power is 2.
  • For variable y, the minimum power is also 2.

Combining these factors, we find the GCF is \(x^2 y^2\). This monomial is the largest common factor present in every term of the polynomial. Once identified, the GCF acts like a common divisor throughout the expression, simplifying the factoring process.

Monomial

A monomial is a single-term algebraic expression that consists of numbers, variables, or the product of numbers and variables. In the context of polynomial expressions, monomials act as building blocks. Understanding the structure of monomials is crucial for factorization tasks.

• Consider the term \(x^2 y^2\): Here, \(x^2\) means \(x\) is multiplied by itself, and \(y^2\) follows the same logic for \(y\).
• In our polynomial \(x^3 y^3 - 2x^3 y^2 + x^2 y^2\), each term is a monomial. They are linked by addition or subtraction to form the full polynomial expression.
• Understanding the makeup of these monomials helps in identifying the GCF, as it provides clues about what can be factored out of the polynomial.

Once you recognize monomials, tasks such as factoring become much more intuitive. By simplifying individual terms, you lay the groundwork for more complex factorization challenges.

Factoring Polynomials

Factoring polynomials involves expressing them as the product of their factors, often simplifying calculations or making further algebraic manipulations possible. After identifying the GCF, the next step in the process is to write the original polynomial as a product involving the GCF.

• Begin by dividing each term of the polynomial by the GCF. This will give you a simpler polynomial that, when multiplied back by the GCF, equals the original polynomial.
• For instance, in the polynomial \(x^3 y^3 - 2x^3 y^2 + x^2 y^2\), each term divided by the GCF \(x^2 y^2\) gives us:
• For \(x^3 y^3\): \((x^3 y^3) / (x^2 y^2) = xy\)
• For \(-2x^3 y^2\): \((-2x^3 y^2) / (x^2 y^2) = -2x\)
• For \(x^2 y^2\): \((x^2 y^2) / (x^2 y^2) = 1\)
• This results in a new, simpler expression \(xy - 2x + 1\), and hence the factorized form of the original polynomial is \(x^2 y^2 (xy - 2x + 1)\).

Factoring polynomials, therefore, simplifies the original expression, making it easier to work with in equations or when further factorization is needed. Understanding each step in the factoring process helps deepen comprehension of algebraic expressions.