Problem 20 - Elementary and Intermediate Algebra

Question

Factor each polynomial. $$ 3 y^{3}+3 y^{2} $$

Step-by-Step Solution

Answer

The factored form is \(3y^2(y + 1)\).

Step 1: Identify Common Factors

First, we need to identify the greatest common factor (GCF) for all the terms in the polynomial \(3y^3 + 3y^2\). Both terms have a common factor of \(3y^2\).

Step 2: Factor Out the Common Factor

Next, factor out \(3y^2\) from each term of the polynomial. This means we divide each term by \(3y^2\) and place the common factor outside the parentheses:\[3y^3 + 3y^2 = 3y^2(y + 1)\]

Step 3: Simplify the Expression

The expression inside the parentheses, \(y + 1\), cannot be factored further as it is already simplified. Thus, the fully factored form of the polynomial is \(3y^2(y + 1)\).

Key Concepts

Greatest Common Factor

When we talk about factoring polynomials, one of the primary concepts is the **Greatest Common Factor (GCF)**. Imagine you have a kitchen and you're baking cookies. You need to find which ingredient is used by most of your recipes. The GCF is similar — it helps identify the largest factor shared by all the terms in an algebraic expression.

To find the GCF of a polynomial like \(3y^3 + 3y^2\), we look for the highest factor that each term has in common. In this example, both terms are divisible by \(3y^2\). This means \(3y^2\) is the GCF, since it's the biggest term possible that equally divides into each part of your polynomial.

• Start by looking at the coefficients (numbers in front of variables). Here, both are 3.
• Next, check the variables. For \(y^3\) and \(y^2\), the smallest power of \(y\) that fits into both is \(y^2\).

Recognizing the GCF in polynomial expressions simplifies the factoring process, making it a critical step in algebra.

Polynomial Expressions

**Polynomial expressions** are like the Swiss Army knife of algebra. These expressions include variables raised to an integer power, multiplied by coefficients. For instance, \(3y^3 + 3y^2\) is a polynomial expression, with terms combined together by addition.

Here's what makes up a polynomial expression:

• **Terms**: Parts of the expression separated by addition or subtraction operators (e.g., \(3y^3\) and \(3y^2\)).
• **Coefficients**: Numbers that multiply the variables (e.g., 3 in each term).
• **Variables**: Symbols like \(y\) that represent numbers.
• **Degree**: The highest power of the variable in the polynomial (here it’s 3, coming from the term \(y^3\)).

Understanding these components is essential. They help you spot opportunities for factoring, simplification, or solving equations.

Algebraic Expressions

**Algebraic expressions** encompass a wider universe than just polynomials. These expressions include numbers, variables, and operations, such as addition, subtraction, multiplication, and division.

Imagine algebraic expressions as phrases formed using different math language ingredients. They can be as simple as a single number or as complex as a polynomial with multiple terms.

• **Contains variables**: Like \(y\) in our example.
• **Operations involved**: Incorporates addition \((+)\), subtraction \((-\)), multiplication \((\times)\), and possibly division \((\div)\).
• **Simplification and factoring**: Tools like identifying common factors (GCF) help transform and solve these expressions.

Grasping algebraic expressions equips you to reason mathematically, much like learning the basic grammar of a new language. Once you learn to navigate these expressions, solving problems becomes much smoother and logical.