Problem 102 - Beginning and Intermediate Algebra

Question

Find the following special products. $$(7 n+2)^{2}$$

Step-by-Step Solution

Answer

The special product for the given expression \((7n + 2)^2\) is \(49n^2 + 28n + 4\).

Step 1: Identify terms in the binomial expression

In the expression \((7n + 2)^2\), we can observe that 'a' is equal to \(7n\) and 'b' is equal to 2.

Step 2: Apply binomial theorem

Now, we will apply the binomial theorem formula \((a + b)^2 = a^2 + 2ab + b^2\): \((7n + 2)^2 = (7n)^2 + 2(7n)(2) + (2)^2\)

Step 3: Find the squares and products

Next, we will find the squares of the terms 'a' and 'b' as well as the term '2ab': \((7n)^2 = 49n^2\) \(2(7n)(2) = 28n\) \((2)^2 = 4\)

Step 4: Plug in the values

Now, plug in the values for the squares and products in the expression: \((7n + 2)^2 = 49n^2 + 28n + 4\) Thus, the special product for the given expression is \(49n^2 + 28n + 4\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool for expanding expressions that are raised to a power. It helps simplify what initially looks complicated. In our example,
we used the formula

• \((a + b)^2 = a^2 + 2ab + b^2\)

to simplify
\((7n + 2)^2\). This method involves identifying the two parts of the binomial – here 'a' is \(7n\) and 'b' is 2.
The Binomial Theorem doesn't just apply to squares, either.

• It's used for higher powers like \((a + b)^3\) which follows a pattern of coefficients from Pascal's triangle.
• These coefficients simplify getting a direct expansion of any binomial raised to a power.
• The theorem is essential in algebraic manipulations and solving equations.

Polynomial Expansion

Polynomial expansion involves taking a simple term like
\((7n + 2)^2\) and expanding it into a longer expression, known as a polynomial.
In our case, the expanded form is

• \(49n^2 + 28n + 4\)

Using the idea of special products helps in the polynomial expansion process by simplifying calculations.
When expanding a binomial square,

• The first term in the expansion is always the square of the first term of the binomial, which gave us \(49n^2\).
• The middle term results from twice the product of both terms of the binomial, yielding \(28n\).
• The last term is the square of the second term, resulting in 4.

This technique is not only useful for squaring binomials but also when dealing with polynomials of higher degree.

Algebraic Expressions

Algebraic expressions are the backbone of algebra. They consist of variables, numbers, and operations that can be combined through addition,
subtraction, multiplication, or division. Consider the expression
\((7n + 2)^2\) as an algebraic expression.

• It combines a variable term \(7n\) with a constant \(2\) using addition inside the parentheses.
• Expanding it into \(49n^2 + 28n + 4\) resolves it into a polynomial.

Working with algebraic expressions requires understanding the operations at their core and how to manipulate them effectively.
The process builds a strong foundation for solving equations, systems of equations, and more complex algebraic tasks. It's crucial to recognize patterns like binomials
and special products when simplifying expressions in algebra, enhancing both speed and accuracy in problem-solving.