Problem 59 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. $$(3 p+5)^{2}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Expression

The exercise requires you to expand the expression \((3p + 5)^2\). This is a binomial squared expression which can be expanded using the binomial formula \((a+b)^2 = a^2 + 2ab + b^2\), where \(a = 3p\) and \(b = 5\).

Step 2: Apply the Binomial Formula

Substitute \(a = 3p\) and \(b = 5\) into the binomial formula. The formula becomes: \((3p)^2 + 2(3p)(5) + 5^2\).

Step 3: Calculate Each Term

- Compute \((3p)^2 = 9p^2\).- Compute \(2(3p)(5) = 30p\).- Compute \(5^2 = 25\).

Step 4: Combine All Terms

Combine the calculated terms: \(9p^2 + 30p + 25\). This is the expanded form of the given expression.

Key Concepts

Understanding the Binomial Theorem

The binomial theorem provides a method for expanding expressions like \((a+b)^n\). When we encounter an expression such as \((3p + 5)^2\), the binomial theorem helps us expand it efficiently without having to multiply the binomial by itself manually. The specific case of \((a+b)^2\) follows the formula:

• \(a^2\)
• \(2ab\)
• \(b^2\)

This means that instead of writing \((a+b) \, (a+b)\) and finding the product term by term, we can use this formula to directly find the expanded form. By substituting \(a = 3p\) and \(b = 5\) into this formula, we quickly calculate the expanded form as \( (3p)^2 + 2 \cdot (3p) \cdot 5 + 5^2 = 9p^2 + 30p + 25\). This method saves time and reduces the risk of error.

Expanding Expressions

Expression expansion is a common problem in algebra that involves breaking down expressions into a simpler or more usable form. With binomials, this usually means taking terms within parentheses and expanding them using the binomial theorem or simple distribution. When expanding a binomial squared, like \((3p+5)^2\), we apply the binomial formula, making it easier to handle complex expressions:

• First, apply the individual terms: \((3p)^2\), which yields \(9p^2\).
• Next, compute the middle term: \(2 \cdot 3p \cdot 5\), which calculates to \(30p\).
• Finally, determine the square of the last term, \(5^2\), which gives us \(25\).

The final expanded expression is simply the sum of these results: \(9p^2 + 30p + 25\). Through practice, this process becomes intuitive and is fundamental to mastering algebra.

Polynomial Operations Made Simple

Polynomial operations are foundational in algebra, encompassing a series of techniques used to simplify, add, subtract, and multiply polynomial expressions. In this exercise, we have a polynomial that is generated by expanding a binomial expression.

• The first step is to identify each individual term in the expression.
• Maneuvering polynomials often involves combining like terms, which are terms sharing the same variables with identical exponents.

In the expression \(9p^2 + 30p + 25\),

• there is a quadratic term \(9p^2\), a linear term \(30p\), and a constant \(25\).

The concept of polynomial operations extends beyond expansion to manipulation and simplification of complex polynomials through operations such as addition, subtraction, and further multiplication, often preparing the expressions for equation solving or graphing. Understanding these operations unlocks the ability to tackle more advanced algebra concepts smoothly.