Problem 78 - Algebra A Combined Function

Question

Factor. $$ x^{2} y^{9}+x^{2} y^{3} $$

Step-by-Step Solution

Answer

The expression factors to \(x^2 y^3 (y^6 + 1)\).

Step 1: Identify Common Factors

Look for common factors in both terms of the expression. Both terms have a factor of \(x^2 y^3\).

Step 2: Factor Out the Common Factor

Take \(x^2 y^3\) out of the expression: \[ x^2 y^9 + x^2 y^3 = x^2 y^3 (y^6 + 1) \]

Step 3: Verify the Factoring

Multiply \(x^2 y^3 (y^6 + 1)\) back out to check the work: \(x^2 y^3 \times y^6 = x^2 y^9\) and \(x^2 y^3 \times 1 = x^2 y^3\). The expression is correctly factored.

Key Concepts

Common Factor

In algebra, discovering a common factor is a crucial step in the process of factoring polynomials. The common factor of a polynomial is a factor that is present in every term of the polynomial expression.

• A common factor might be a single number, a variable, or a variable raised to an exponent.
• To identify it, look at each term of the polynomial and determine what numbers or variables they all share.

The common factor simplifies expressions and breaking down polynomial terms becomes easier. In this case, the expression is composed of the terms \(x^2y^9\) and \(x^2y^3\), both sharing the common factor \(x^2y^3\).
By factoring out \(x^2y^3\), you essentially reverse the distribution process, dividing each term by the common factor, which results in a more simplified expression. Finding common factors helps not only in simplification but also in understanding the structure of the polynomial thoroughly.

Polynomial Terms

Polynomial terms are the building blocks of polynomials and consist of products or multiplications of variables, coefficients, and exponents. Each term in a polynomial can be seen as a separate entity within the expression until they become connected through addition or subtraction.
For example, in the polynomial \(x^2y^9 + x^2y^3\), there are two terms that make up the entire expression:

• \(x^2y^9\)
• \(x^2y^3\)

Understanding the concept of polynomial terms is vital for efficient factoring since recognizing how terms relate to each other can simplify operations like addition, subtraction, or factoring itself. On analyzing the expression, we notice that although each term involves different powers of variables, they both share a common factor, facilitating the process of combining and simplifying them. Recognizing and manipulating these terms effectively is key to solving complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers connected through arithmetic operations. They form the foundation of algebra, representing mathematical phrases that can encompass simpler or more complex elements like constants, coefficients, variables, and polynomial terms.
In exercises involving expressions like \(x^2y^9 + x^2y^3\), factoring is an essential skill. It involves breaking down a more complicated expression into a product of simpler factors that are easier to understand and use in further calculations.

• The primary goal is to simplify expressions which leads to easier computations.
• Using different properties and rules, such as distributive property or identifying common factors, plays a critical role in managing these expressions.

When working with algebraic expressions, the ability to factor them correctly aids in solving equations and inequalities, allowing you to find solutions more efficiently. Understanding this concept enriches your algebraic toolkit, enabling you to handle various mathematics problems with skill and confidence.