Problem 85 - College Algebra with Modeling and Visualization

Question

Multiply the expressions. $$(x+4)^{2}$$

Step-by-Step Solution

Answer

The expanded expression is \(x^2 + 8x + 16\).

Step 1: Understanding the Expression

We are given the expression \((x+4)^2\). This expression means that we have to multiply \((x+4)\) by itself.

Step 2: Apply the Formula for Squaring a Binomial

The expression \((x+4)^2\) can be expanded using the binomial square formula: \[(a+b)^2 = a^2 + 2ab + b^2.\] In our case, \(a = x\) and \(b = 4\).

Step 3: Calculate Each Term

Substitute \(a = x\) and \(b = 4\) into the binomial formula: - First term: \(a^2 = x^2\).- Second term: \(2ab = 2 \cdot x \cdot 4 = 8x\).- Third term: \(b^2 = 4^2 = 16\).

Step 4: Sum the Terms

Add each of the terms obtained in step 3 to get the fully expanded expression: \[x^2 + 8x + 16.\]

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics that allow us to represent numbers and operations in a concise form. They consist of variables, constants, and arithmetic operations such as addition and multiplication.
In the exercise, the expression \((x+4)^2\) is an example of an algebraic expression. Here, "\(x\)" is a variable that can represent any number, and "4" is a constant.

• Variables: Symbols like \(x\) that stand for unknown values.
• Constants: Specific numbers like "4" that don’t change.
• Operations: Actions such as squaring and multiplying.

By knowing how to manipulate algebraic expressions, we can solve equations, analyze mathematical relationships, and understand functional relationships.

Polynomial Multiplication

Polynomial multiplication involves combining two or more polynomials to form a new polynomial. In our exercise, multiplying \((x+4)\) by itself is a specific case of polynomial multiplication known as squaring a binomial.
To perform this operation, we use distributive properties and organize the results:

• Squaring: \((a+b)^2\) means multiplying \((a+b)\) by itself.
• Distributive Property: Apply the multiplication across each expression, like \(a\) times \(b\).
• Combining Like Terms: Simplify the expression by merging similar terms.

With polynomials, it's key to track each term and add them properly to reach the expanded form. This ensures an accurate depiction of the algebraic identity being expressed.

Quadratic Expressions

Quadratic expressions, often in the form of \(ax^2 + bx + c\), play an essential role in algebra and can describe various phenomena. When squaring the binomial \((x+4)^2\), the resulting polynomial \(x^2 + 8x + 16\) is quadratic.
Quadratic expressions are characterized by:

• Second Degree: The highest power of the variable \(x\) is \(x^2\).
• Three Terms: Typically represented by a three-term polynomial.
• Vertex Form: Often manipulated for analyzing parabolas.

Understanding quadratics is crucial for solving quadratic equations, graphing parabolas, and modeling situations such as projectile motion in physics. By mastering this structure, one can predict and design with greater precision.