Problem 88 - College Algebra with Modeling and Visualization

Question

Multiply the expressions. $$(3 x+5)^{2}$$

Step-by-Step Solution

Answer

\((3x + 5)^2 = 9x^2 + 30x + 25\).

Step 1: Identify the Expression Form

The expression \((3x + 5)^2\) is in the form of \((a + b)^2\). This indicates that we can use the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).

Step 2: Apply the Formula

Using the formula, we let \(a = 3x\) and \(b = 5\), and substitute these values into the formula \((a + b)^2 = a^2 + 2ab + b^2\). This leads to: \((3x + 5)^2 = (3x)^2 + 2(3x)(5) + (5)^2\).

Step 3: Calculate Each Term

Calculate each term separately:1. \((3x)^2 = 9x^2\).2. \(2(3x)(5) = 30x\).3. \((5)^2 = 25\).

Step 4: Combine the Terms

Combine all calculated terms to get the final expression: \(9x^2 + 30x + 25\).

Key Concepts

Square of a Binomial

The square of a binomial expression is a fundamental concept in algebra. A binomial is an algebraic expression that contains exactly two terms. When you square a binomial, you multiply it by itself. For example, squaring the binomial \((a + b)\) involves computing \((a + b) \times (a + b)\).

To simplify the process, we use a special formula:

• \((a + b)^2 = a^2 + 2ab + b^2\)

This formula helps you quickly find the square without doing all the multiplication. It's derived from applying the distributive property to expand the binomial.

So, for \((3x + 5)^2\), let \(a = 3x\) and \(b = 5\). The formula breaks down the squaring process into manageable parts. Calculate \(a^2 = (3x)^2\), \(2ab = 2(3x)(5)\), and \(b^2 = 5^2\). The solution: \(9x^2 + 30x + 25\).

Polynomial Multiplication

Polynomial multiplication is an essential skill for expanding and simplifying expressions. When we multiply polynomials, we use the distributive property, which states that each term in the first polynomial must be multiplied by each term in the second polynomial.

In the context of squaring a binomial, polynomial multiplication involves distributing each term of the binomial against itself. Let's consider the general expression \((a + b)\).

To expand \((a + b)^2\) using multiplication, follow these steps:

• First, multiply \(a\) by \(a\), giving \(a^2\).
• Next, multiply \(a\) by \(b\), giving \(ab\).
• Then, multiply \(b\) by \(a\), resulting in another \(ab\).
• Finally, multiply \(b\) by \(b\), giving \(b^2\).

Combine these results: \(a^2 + ab + ab + b^2 = a^2 + 2ab + b^2\). Hence, rewriting \((3x+5)^2\) involves organizing terms and calculating values, as outlined in the exercise solution.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations like addition, subtraction, multiplication, and division. They are powerful tools in mathematics that help model real-world problems. Understanding algebraic expressions is fundamental to solving equations and manipulating algebraic forms.

In our problem, \((3x + 5)^2\), the expression inside the parentheses is an algebraic expression composed of two terms: \(3x\) and \(5\). Each term represents a specific part of the expression:

• \(3x\): A variable term where \(3\) is a coefficient, and \(x\) is the variable.
• \(5\): A constant term.

When you square a binomial like this, you don't just increase the power but transform it into a polynomial with more terms by applying the binomial square expansion method.

By recognizing and operating on these expressions, one can transform and simplify complicated expressions, aiding in more complex mathematical problem-solving. This exercise helps illustrate the principles and utility of managing algebraic expressions effectively.