Problem 41 - College Algebra

Question

Find each product. $$(x+2)^{2}$$

Step-by-Step Solution

Answer

The product of \((x+2)^{2}\) is \(x^2 + 4x + 4\).

Step 1: Expand the binomial

Rewrite \((x+2)^{2}\) as \((x+2)\) times \((x+2)\) himself. So it becomes \((x+2)\) * \((x+2)\).

Step 2: Apply the distributive property

Multiply each term in the first bracket by each term in the second bracket. This results in \(x*x + 2*x + x*2 + 2*2\).

Step 3: Simplify

Simplify the expressions which results in \(x^2 + 2x + 2x + 4\).

Step 4: Combine like terms

Combine the like terms to get the final answer \(x^2 + 4x + 4\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term across terms inside a parenthesis. It's used to simplify expressions and is essential when dealing with operations inside brackets.
For example, when you have an expression such as \(a(b+c)\), the distributive property tells us that this can be expanded to \(ab+ac\).
In the context of the given problem, \((x+2)^2\), it means expanding by multiplying \(x+2\) by itself, \((x+2) \, * \,(x+2)\).
To apply this property, you distribute each term from the first bracket across every term of the second bracket.

• \(x \cdot x = x^2\)
• \(x \cdot 2 = 2x\)
• \(2 \cdot x = 2x\)
• \(2 \cdot 2 = 4\)

By following this step-by-step multiplication, you correctly apply the distributive property, which is key to expanding situations involving polynomials, like binomials.

Polynomials

Polynomials are expressions made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They are a fundamental concept in algebra and can take various shapes and degrees.
In our example, the expression \(x^2 + 4x + 4\) is a polynomial. It consists of three terms:

• \(x^2\) is the quadratic term since it involves the variable raised to the power of 2.
• \(4x\) is the linear term, where the variable is raised to the first power.
• \(4\) is the constant term, not involving the variable.

Understanding polynomials helps with graphing, factoring, and solving equations. In higher degrees, polynomials can become more complex, but their foundation remains rooted in these basic operations and structure, as seen in \(x^2 + 4x + 4\).

Algebraic Expressions

Algebraic expressions combine numbers, variables, and arithmetic operations to represent a value or a set of values. These expressions are the building blocks of algebra that allow for solving equations and modeling real-world problems.
The expression \(x^2 + 4x + 4\) from the exercise is an algebraic expression that showcases how variables and numbers interact through addition and multiplication.

• Variables like \(x\) represent unknown values that can change.
• Constants like \(4\) are fixed values.
• Operators such as \(+\) and \(-\) join different parts of an expression.

The process of working with algebraic expressions often involves simplifying, factoring, and solving them to find the value of the variables. Grasping how these expressions are formed and manipulated forms the foundation of more advanced mathematical problem-solving.