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Chapter 0: Problem 48

Perform the indicated operations and simplify. $$ (7 v-2)^{2} $$

Short Answer

Expert verified
49v^{2} - 28v + 4

Step by step solution

01

Write down the binomial formula

Recall the binomial formula for squaring a binomial: (a - b)^{2} = a^{2} - 2ab + b^{2}
02

Identify the terms

Identify the terms in the given expression a = 7v b = 2
03

Apply the binomial formula

Substitute the identified terms into the binomial formula and simplify: (7v-2)^{2} = (7v)^{2} - 2 * 7v * 2 + (2)^{2}
04

Simplify each term

Simplify the terms obtained: (7v)^{2} = 49v^{2} 2 * 7v * 2 = 28v (2)^{2} = 4 When substituting back, we get: 49v^{2} - 28v + 4
05

Combine and write the final expression

Write the simplified expression obtained: 49v^{2} - 28v + 4

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Key Concepts

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

Binomial Formula
When dealing with algebraic expressions, especially binomials, the binomial formula is essential. The binomial formula helps in expanding expressions that are squared or raised to higher powers. For example, the binomial formula for squaring is given by: \[(a - b)^{2} = a^{2} - 2ab + b^{2} \].
In our exercise, the expression is \((7v-2)^{2}\). By identifying \(a\) as \(7v\) and \(b\) as \(2\), we can use the binomial formula to expand it.
Applying the binomial formula correctly simplifies complex algebraic operations and allows us to break down expressions into manageable steps.
Polynomial Simplification
Simplifying polynomials involves combining like terms and performing basic arithmetic operations. Once we apply the binomial formula to \((7v-2)^{2}\), we get: \((7v)^{2} - 2 * 7v * 2 + 2^{2}\).
To simplify further: \((7v)^{2} = 49v^{2}\), \(-2 * 7v * 2 = -28v\), and \(2^{2} = 4\).
Putting these terms together results in the polynomial \((49v^{2} - 28v + 4)\).
Note that each polynomial term should be combined by performing operations such as addition or subtraction. This allows us to express the polynomial in its simplest form.
Algebraic Expressions
Algebraic expressions involve variables, coefficients, and constants. Understanding how to manipulate and simplify these expressions is crucial in algebra.
In the given exercise, the expression \((7v-2)^{2}\) involves a variable \(v\) and constants \(7\) and \(2\).
Using rules like the binomial formula and simplifying polynomials, we break down the expression into easier parts before recombining them.
This process not only helps simplify the original problem but also aids in developing a deeper understanding of algebraic operations and the relationships between different components of the expressions.

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Most popular questions from this chapter

Simplify each expression. $$ (6 x+5)(6 x-5)-(6 x+5)^{2} $$

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We know that \((a-b)^{2}=a^{2}-2 a b+b^{2}\). Derive a special product formula for \((a-b)^{3}\).
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