Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 10 x y+15 x y^{2} $$

Step-by-Step Solution

Answer

The polynomial factors to \(5xy(2 + 3y)\).

Step 1: Identify the Common Factor

Look at each term in the polynomial and find the greatest common factor (GCF). The terms in the polynomial are \(10xy\) and \(15xy^2\). The factors of \(10xy\) are \(10\), \(x\), and \(y\), and the factors of \(15xy^2\) are \(15\), \(x\), \(y\), and \(y\). The common factors between both terms are \(5\), \(x\), and \(y\). Thus, the GCF is \(5xy\).

Step 2: Factor Out the Common Factor

Factor out the GCF, \(5xy\), from the polynomial. Divide each term by \(5xy\) and write them inside the parenthesis. This gives us \[5xy (2 + 3y).\]

Step 3: Express the Factored Form

Combine the factors outside and inside the parenthesis to express the fully factored form of the polynomial. The polynomial \(10xy + 15xy^2\) factors to \(5xy(2 + 3y)\).

Key Concepts

Understanding the Greatest Common Factor (GCF)

The greatest common factor (GCF) is a key component in factoring polynomials. It's the largest factor that divides two or more numbers. In the context of polynomials, the GCF includes variables as well.
To find the GCF of a polynomial's terms, follow these steps:

• List all factors of each term. Include coefficients and variables.
• Identify the common factors across each term.
• Choose the largest common factor.

For example, in the polynomial terms \(10xy\) and \(15xy^2\), their factors include numbers and variables:

• **For \(10xy\):** 10, x, y
• **For \(15xy^2\):** 15, x, y, y

The common factors here are \(5, x,\) and \(y\). So, the GCF is \(5xy\). Highlighting these key steps ensures you're equipped to tackle various polynomials by identifying and using the GCF effectively.

Transition to Factored Form

Factoring a polynomial essentially means breaking it down into simpler components. The factored form usually has a product of its greatest common factor and a simpler polynomial. This form is concise and allows better manipulation of the expression.
To express a polynomial in factored form:

• Identify the greatest common factor.
• Divide every term by the GCF.
• Express the division result within parentheses, multiplying by the GCF.

In our exercise, the polynomial \(10xy + 15xy^2\) simplifies using its GCF. Dividing each term by \(5xy\), we get:

• \(\frac{10xy}{5xy} = 2\)
• \(\frac{15xy^2}{5xy} = 3y\)

Put together, the factored form is \(5xy(2 + 3y)\). Such simplification aids in solving equations and understanding algebraic structures.

Exploring Polynomials

Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. These algebraic structures can incorporate different operations and can be represented in standard or factored form.
In a polynomial, each component such as \(x, y\) or their powers (\(x^2, y^2\), etc.) is referred to as a "term." Importantly, the power of a variable determines the "degree" of the term.
With polynomials, operations like factoring, expanding, and simplifying are crucial for solving equations and analyzing functions.

• **Monomials:** Single terms like \(3x\) or \(-5y^2\).
• **Binomials:** Two terms, as seen in \(x + y\).
• **Trinomials:** Three terms, like \(x^2 + 3x + 2\).

Understanding how polynomials are structured and function is fundamental in algebra. Factoring them, like in our problem, is about reorganizing them into useful forms for practical computations and theoretical explorations.