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Chapter 0: Problem 53

Perform the indicated operations and simplify. $$ [(u+v)-w][(u+v)+w] $$

Short Answer

Expert verified
(u+v)^2 - w^2 = u^2 + 2uv + v^2 - w^2

Step by step solution

01

Recognize the Expression

Identify that the expression \([(u+v)-w][(u+v)+w]\) is in the form of \[ (a - b)(a + b) \].
02

Apply the Difference of Squares Formula

Use the difference of squares formula, which states \[ (a - b)(a + b) = a^2 - b^2 \]. Here, \( a = u + v \) and \( b = w \).
03

Substitute and Simplify

Substitute the identified \( a \) and \( b \) into the formula: \[ (u+v)^2 - w^2 \].
04

Expand the Squared Terms

Expand \( (u+v)^2 \) to get \[ u^2 + 2uv + v^2 \]. This gives us the expression \[ u^2 + 2uv + v^2 - w^2 \].

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Key Concepts

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

Difference of Squares Formula
When you hear the term 'Difference of Squares', it refers to a specific algebraic expression. In our problem, we're dealing with an expression that can be written as \((a-b)(a+b)\). The Difference of Squares formula states that \((a-b)(a+b) = a^2 - b^2\). This formula turns the product of two binomials into a simple subtraction of squares.

This concept is useful because it simplifies complex expressions into something much easier to handle.
Polynomial Expansion
Polynomial Expansion involves expanding expressions that are multiplied together. In our exercise, we expanded \((u+v)^2\).

To expand this, you use the distributive property. This means you multiply each term in \(u+v\) by each term in the other \(u+v\). This results in \(u^2 + 2uv + v^2\).

Understanding polynomial expansion helps in breaking down more complicated algebraic expressions into simpler parts.
  • \(u + v \implies u^2, uv, v^2 \)
  • Add the middle term twice: \uv + uv\ to get \2uv\.
Algebraic Expressions
An algebraic expression is made up of variables, numbers, and operations. Examples include \(u + v\) and \(w^2\).

In our exercise, we started with the expression \([(u+v)-w][(u+v)+w]\). We identified parts of this expression as \(u + v\) and \(w\).

Algebraic expressions can often be simplified using the Difference of Squares formula. This simplifies them for easier manipulation and understanding.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves reducing them to their simplest form. In our problem, after applying the Difference of Squares formula, we turned our expression into \(u^2+2uv+v^2-w^2\).

This expression is much simpler and easier to understand than the original. Simplifying often involves combining like terms or using formulas like the Difference of Squares.
  • Start with identifying terms and their roles.
  • Apply known formulas (e.g., Difference of Squares).
  • Combine like terms and simplify.

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Most popular questions from this chapter

Determine if the statement is true or false. The product of two polynomials each of degree 4 will be less than degree 8 .

Write an inequality representing the given statement. \(b\) is positive.

Suppose that \(x\) represents the larger of two consecutive odd integers. a. Write a polynomial that represents the smaller integer. b. Write a polynomial that represents the sum of the two integers. Then simplify. c. Write a polynomial that represents the product of the two integers. Then simplify. d. Write a polynomial that represents the difference of the squares of the two integers. Then simplify.

Multiply and simplify. $$ (2 x-5)(x+4) $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ \left(5 a^{2} \sqrt{b}+7 b^{2} \sqrt{a}\right)^{2} $$
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