Problem 41 - A Graphical Approach to College Algebra

Question

Find each product. $$(4 m+2 n)^{2}$$

Step-by-Step Solution

Answer

\((4m + 2n)^2 = 16m^2 + 16mn + 4n^2\).

Step 1: Identify the Formula

To find the product of the expression \((4m + 2n)^2\), we can apply the binomial square formula. The formula is \((a + b)^2 = a^2 + 2ab + b^2\). For this expression, \(a = 4m\) and \(b = 2n\).

Step 2: Apply the Formula

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), substitute \(a = 4m\) and \(b = 2n\). Compute each term:- \(a^2 = (4m)^2 = 16m^2\)- \(b^2 = (2n)^2 = 4n^2\)- \(2ab = 2 \times 4m \times 2n = 16mn\)

Step 3: Combine the Terms

Combine the terms obtained from the application of the formula:\((4m + 2n)^2 = 16m^2 + 16mn + 4n^2\)

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols can represent numbers, functions, or other mathematical entities. The goal of algebra is to solve equations and find the values of unknown variables. It provides a powerful framework to express real-world problems in mathematical form.
Algebra forms the foundation for many advanced mathematical concepts. One of its key features is the use of variables, which allows us to work with generalizations instead of specific numbers. This abstraction enables us to create formulas that can solve a wide range of problems.
When studying algebra, understanding how to manipulate expressions is essential. You'll often work on tasks like expanding, factoring, and simplifying polynomial expressions. These tasks involve applying specific mathematical rules and properties, such as the distributive property and the commutative property. In this context, the binomial theorem is a valuable tool that helps in expanding expressions raised to a power.

Polynomial Expansion

Polynomial expansion refers to the process of expressing a polynomial raised to a power as a sum of its terms. The binomial theorem is a key principle used in polynomial expansion. This theorem offers a quick way to expand expressions of the form \((a + b)^n\).
A binomial can be expanded using the formula \((a + b)^2 = a^2 + 2ab + b^2\). This formula derives from the process known as 'foiling,' which involves multiplying two binomials together: each component of one polynomial is multiplied by each component of another, and the terms are then summed.

• First: Multiply the first terms of each binomial.
• Outside: Multiply the outer terms of the binomials.
• Inside: Multiply the inner terms of the binomials.
• Last: Multiply the last terms of each binomial.

In essence, polynomial expansion turns a relatively simple algebraic expression into a longer, more detailed expression that offers insight into the behavior of the polynomial. This process is essential for solving many types of algebraic problems, providing a clear pathway to analyze complex expressions.

Quadratic Expressions

Quadratic expressions are algebraic expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. These expressions are vital in mathematics as they frequently describe parabolic relations.
In this age of algebra, understanding how to work with quadratic expressions is crucial. They appear commonly in problems involving the motion of objects, optimization problems in calculus, and in various engineering fields. A specific type of quadratic expression is a perfect square trinomial, which results from squaring a binomial.

For example, if you square \((x + y)\), you end up with the trinomial \(x^2 + 2xy + y^2\), representing the binomial squared.

An expression like \((4m + 2n)^2\) is squared to yield \(16m^2 + 16mn + 4n^2\), as detailed in the steps above. This reveals the quadratic structure embedded within binomial expansions.

Quadratic expressions can be factored back into binomials, allowing for solutions of quadratic equations. This is often solved using methods like completing the square or applying the quadratic formula, each providing critical insights into the nature and roots of the expression.