Problem 27 - College Algebra

Question

For the following exercises, expand the binomial. $$(4 p+9)^{2}$$

Step-by-Step Solution

Answer

The expanded form of \((4p + 9)^2\) is \(16p^2 + 72p + 81\).

Step 1: Recognize the Binomial

The expression we need to expand is \((4p + 9)^2\). This is a binomial expression raised to the power of 2.

Step 2: Apply the Binomial Theorem

When expanding a binomial of the form \((a + b)^n\) where \(n = 2\), use the formula: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 4p\) and \(b = 9\).

Step 3: Calculate \(a^2\)

Calculate \((4p)^2\). \((4p)^2 = 16p^2\).This is the first term of the expansion.

Step 4: Calculate \(2ab\)

Calculate \(2 \cdot (4p) \cdot 9\).\(2 \cdot 4p \cdot 9 = 72p\).This is the second term of the expansion.

Step 5: Calculate \(b^2\)

Calculate \(9^2\).\(9^2 = 81\).This is the third term of the expansion.

Step 6: Write the Expanded Form

Combine all terms from the previous steps: \(16p^2 + 72p + 81\). This is the expanded form of the binomial \((4p + 9)^2\).

Key Concepts

Binomial Theorem

The Binomial Theorem provides a powerful tool for expanding polynomial expressions. It tells us how to expand expressions that look like \((a + b)^n\). This theorem is particularly simple to use when \(n = 2\), which is a very common case.In our problem, you can see the expression \((4p + 9)^2\).This is a binomial raised to a power. Binomials are simply expressions with two terms, in this case, \(4p\) and \(9\). To expand using the Binomial Theorem when \(n = 2\), we apply the formula \((a + b)^2 = a^2 + 2ab + b^2\). This means we can break the problem into smaller steps that are easier to handle:

• Identify \(a\) and \(b\) in the binomial. Here \(a = 4p\) and \(b = 9\).
• Calculate each part of the expansion: \(a^2\), \(2ab\), and \(b^2\).
• Combine these calculations into the expanded form.

Polynomial Expressions

Polynomial expressions are combinations of numbers and variables raised to whole-number exponents. They include terms that are connected by plus or minus signs. The expression \((4p + 9)^2\) is not yet a polynomial until fully expanded.
When we expanded our binomial using the Binomial Theorem, we created a polynomial: \(16p^2 + 72p + 81\). This expression has three terms connected by addition:

• \(16p^2\): A term with the variable \(p\) raised to the second power.
• \(72p\): This term contains the variable \(p\) raised to the first power.
• \(81\): A constant term without any variable.

Recognizing each term is key when understanding and working with polynomials. Each part plays a unique role in polynomial calculations, where powers of variables come into play.

Algebraic Manipulation

Algebraic manipulation involves using various techniques to simplify or rearrange expressions. When expanding a binomial like \((4p + 9)^2\), applying algebraic manipulation is essential. It involves a logical sequence of operations to transform one form into another.
The steps involved require:

• Careful calculation of each required result such as \(a^2\), \(2ab\), and \(b^2\).
• Simplification of these components.
• Finally, combining them to create a single polynomial expression.

For the expression \( (4p + 9)^2 \), calculate:

• \((4p)^2 = 16p^2\) by squaring the first term.
• \(2 \cdot 4p \cdot 9 = 72p\) by multiplying and simplifying the middle part.
• \(9^2 = 81\) by squaring the last term.
• The last step is adding these results together: \(16p^2 + 72p + 81\).

Understanding these steps of algebraic manipulation is fundamental for dealing with complex expressions and ensures accuracy in solutions.