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Chapter 7: Problem 82

Perform the indicated operations. Simplify, if possible. $$ (7 m-2 n)^{2} $$

Short Answer

Expert verified
The simplified form is \( 49m^2 - 28mn + 4n^2 \).

Step by step solution

01

Understand the Expression

The given expression is \( (7m - 2n)^2 \). This involves squaring a binomial. Recall that \( (a - b)^2 = a^2 - 2ab + b^2 \).
02

Identify the Components

In the expression \( (7m - 2n) \), identify \( a = 7m \) and \( b = 2n \). This will help apply the binomial square formula correctly.
03

Apply the Binomial Square Formula

Use the formula \( (a-b)^2 = a^2 - 2ab + b^2 \) to expand the expression. Substitute \( a = 7m \) and \( b = 2n \) into the formula: \( (7m - 2n)^2 = (7m)^2 - 2(7m)(2n) + (2n)^2 \).
04

Calculate Each Term Individually

Calculate \( (7m)^2 = 49m^2 \). Calculate \( -2(7m)(2n) = -28mn \). Calculate \( (2n)^2 = 4n^2 \).
05

Combine the Results

Combine all the calculated terms to get the expanded and simplified expression: \( 49m^2 - 28mn + 4n^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic simplification
Algebraic simplification involves breaking down complex expressions into simpler terms. It requires combining like terms, applying distributive properties, and reducing expressions wherever possible. When you expand (7m 鈥� 2n)^2 to get 49m^2 鈥� 28mn + 4n^2, you are simplifying the binomial expression to a polynomial. Combining terms and applying mathematical operations carefully is crucial. This simplification allows you to better understand and use the expression in different algebraic operations.
Polynomials
A polynomial is an algebraic expression that consists of variables and coefficients, arranged in terms raised to non-negative integer powers. In our example, expanding \(7m - 2n)^2\) results in the polynomial \(49m^2 - 28mn + 4n^2\). Understanding polynomials is essential as they are the foundation for many algebraic operations, from factoring to solving equations. Breaking down polynomials into simpler parts, like monomials and binomials, makes understanding and working with them easier. Polynomials can also be added, subtracted, multiplied, or divided, making them extremely versatile in algebra.

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Most popular questions from this chapter

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