Problem 21 - Elementary and Intermediate Algebra

Question

Factor each polynomial. $$ 15 x^{2} y-10 x^{2} y^{2} $$

Step-by-Step Solution

Answer

The polynomial factors to \(5x^2y(3 - 2y)\).

Step 1: Identify Common Factors

Look at the terms of the polynomial: \(15x^2y\) and \(-10x^2y^2\). Notice that both terms have common factors. Each term contains \(x^2y\). The numerical coefficients 15 and 10 can both be divided by 5. Therefore, the greatest common factor (GCF) is \(5x^2y\).

Step 2: Factor Out the Greatest Common Factor

Divide each term in the polynomial by the greatest common factor \(5x^2y\):- \(\frac{15x^2y}{5x^2y} = 3\)- \(\frac{-10x^2y^2}{5x^2y} = -2y\)This means we can express the polynomial as: \(5x^2y(3 - 2y)\).

Step 3: Write the Fully Factored Form

The polynomial \(15x^2y - 10x^2y^2\) is now written as the product of \(5x^2y\) and \(3 - 2y\). Therefore, the fully factored form is \(5x^2y(3 - 2y)\).

Key Concepts

The Greatest Common Factor (GCF) in Polynomials

Factoring polynomials often begins with finding the greatest common factor, or GCF. This is essentially the largest factor that divides each term individually in the polynomial. In our example, we have two terms: \(15x^2y\) and \(-10x^2y^2\).- **Identifying the GCF**: Look for common variables and their lowest power appearing in each term. - Here, both terms include \(x^2y\) as a common variable factor.- **Numerical GCF**: Check the coefficients (numerical parts) of each term. - For 15 and 10, the greatest number that divides both is 5.Thus, the GCF for this polynomial is the product of the numerical GCF and the common variable factor, which results in \(5x^2y\).
This process simplifies the polynomial and prepares it for polynomial division. Once you have the GCF, you can factor it out from each term, leading us further into the factoring process.

Basics of Polynomial Division

Polynomial division is a pivotal concept when dealing with expressions that have a common factor. This technique helps to simplify expressions, making them easier to handle and understand. Here's how it works using our example:- **Divide Each Term by the GCF**: Begin by dividing each term by the greatest common factor \(5x^2y\). - For \(15x^2y\), the result of dividing by \(5x^2y\) is 3. - For \(-10x^2y^2\), the division by \(5x^2y\) simplifies to \(-2y\).Performing these divisions helps to break down the original polynomial, giving a simpler expression. Polynomial division is a crucial step in obtaining a clearer view of the expression, ultimately leading you to the simpler factored form. This calculation aids in deeper insights into the polynomial's structure and components.

Understanding the Factored Form

Once you have employed the GCF and polynomial division, what remains is writing the polynomial in its factored form. The factored form is a product of two or more simpler expressions that, when multiplied together, give the original polynomial.For our situation, after factoring out \(5x^2y\), the polynomial becomes:- \(5x^2y(3 - 2y)\)This expression breaks the polynomial down into its essential components:- The term \(5x^2y\) is the GCF, representing the shared elements in the terms.- The binomial \(3 - 2y\) is what's left after division, highlighting the difference between the terms.The beauty of the factored form lies in its ability to simplify calculations and solutions, especially in more complex scenarios. Understanding and practicing this form allows for more insight into polynomials and easier manipulation in algebraic equations.