Problem 50 - A Graphical Approach to College Algebra

Question

Find each product. $$[(2 m+7)-n]^{2}$$

Step-by-Step Solution

Answer

The expanded form is \(4m^2 + 28m + 49 - 4mn - 14n + n^2\).

Step 1: Understand the Expression

The given expression \([(2m + 7) - n]^2\) is a squared expression, which means you have to multiply it by itself.

Step 2: Apply the Binomial Theorem

To expand \([(2m + 7) - n]^2\), apply the formula \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 2m + 7\) and \(b = n\).

Step 3: Compute Each Term

First, calculate \((2m + 7)^2\):\[(2m + 7)^2 = (2m)^2 + 2 \cdot (2m) \cdot 7 + 7^2 = 4m^2 + 28m + 49\].

Step 4: Compute the Middle Term

Next, compute \(-2 \times (2m + 7) \times n\):\[-2 \times (2m + 7) \times n = -2(2mn + 7n) = -4mn - 14n\].

Step 5: Compute the Square of the Second Term

Finally, compute \(n^2\):\[n^2 = n^2\].

Step 6: Combine All Terms

Put together all terms calculated:\[4m^2 + 28m + 49 - 4mn - 14n + n^2\].This gives the expanded form of the original expression.

Key Concepts

Algebraic Expressions

Algebraic expressions form the building blocks of algebra and consist of numbers, variables, and arithmetic operations (+, -, *, /). In the expression \((2m + 7) - n\), we see an example of an algebraic expression made up of terms, where "term" denotes a component of the expression combined by mathematical operations.
In our expression, **\((2m + 7)\)** is grouped together and signifies a binomial expression—an algebraic expression with two distinct terms. The subtraction of \(n\) makes it a trinomial when combined.

It's crucial to remember that algebraic expressions can't be solved like equations, but they can be simplified or manipulated. Operations such as addition and multiplication are used to combine terms, and together with parentheses, they define the hierarchy of operations that structure the expression. Recognizing these components is the first step in understanding how to expand the expression.

Polynomial Expansion

Polynomial expansion is the process of spreading out an expression consisting of multiple terms into a more detailed, expanded form. In expanding a polynomial, we take a compact polynomial and write it as a sum of simpler terms.
In the given problem, **applying the binomial theorem** simplifies the expansion of \([(2m + 7) - n]^2\). This theorem provides a straightforward method to expand binomials raised to any positive integer. By using the formula \((a - b)^2 = a^2 - 2ab + b^2\), we see how to expand and express each part of the binomial.

The steps are as follows:

• Calculate \(a^2\): Expand \((2m + 7)^2\) using multiplication: \(4m^2 + 28m + 49\).
• Find \(-2ab\): Multiply each component of the binomial by \(-2n\) to yield \(-4mn - 14n\).
• Determine \(b^2\): As \(b = n\), squaring \(n\) results in \(n^2\).

Each step follows the binomial theorem closely, ensuring that all possible products between terms are calculated and combined to achieve the expanded expression.

Exponents

Exponents are a way of expressing repeated multiplication in a compact form. When you see \(x^2\), it signifies \(x\) multiplied by itself. Understanding exponents is essential for expanding polynomials because they show the number of times a term is multiplied by itself.
The problem we solve uses exponents to indicate the square of the expression \([(2m + 7) - n]\). The exponent \(^2\) tells us to multiply \((2m + 7) - n\) by itself. This squaring process requires careful application of distributive properties to ensure every term interacts with others properly.

Key aspects of handling exponents in this context include:

• Simplified Notation: Instead of writing out repetitive multiplication, exponents provide a simpler notation.
• Distributive Property: Exponents distribute over addition/subtraction—a core principle used in the expansion process.
• Result of Squaring: When squaring, every term gets squared individually, as shown in the complete expansion \(4m^2 + 28m + 49 - 4mn - 14n + n^2\).

Recognizing how to work with exponents helps streamline the calculation and ensures precision in expressions, especially when multiple terms and variables are involved.