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Chapter 1: Problem 83

Factor the expression completely. $$(a+b)^{2}-(a-b)^{2}$$

Short Answer

Expert verified
The expression factors to \(4ab\).

Step by step solution

01

Recognize the Algebraic Identity

The given expression is \[(a+b)^2 - (a-b)^2.\] This is a difference of squares, which can be factored using the identity: \[x^2 - y^2 = (x+y)(x-y).\] In the given expression, set \(x = (a+b)\) and \(y = (a-b)\).
02

Apply the Difference of Squares

Using the identity, we apply it to the expression:\[(a+b)^2 - (a-b)^2 = [(a+b) + (a-b)][(a+b)-(a-b)].\]
03

Simplify Each Term

Simplify the first term of the product:\[(a+b) + (a-b) = a + b + a - b = 2a.\]Now simplify the second term:\[(a+b) - (a-b) = a + b - a + b = 2b.\]
04

Write the Final Factored Form

With the simplified expressions, write the product:\[(a+b)^2 - (a-b)^2 = (2a)(2b).\]This simplifies to:\[4ab.\]

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Key Concepts

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

Difference of Squares
The difference of squares is a valuable algebraic identity used in many math problems. It is represented by the formula \(x^2 - y^2 = (x+y)(x-y)\). This identity is useful because it allows us to break down expressions involving the squares of terms, simplifying complex problems.
When you see an expression like \((a+b)^2 - (a-b)^2\), it is an opportunity to use the difference of squares formula. This expression fits perfectly because it includes two squared terms separated by a subtraction sign.
Recognizing these patterns can help in transforming the problem into a simpler one, making calculations quicker and easier.
Factoring Expressions
Factoring expressions is a key skill in algebra. It refers to writing a polynomial as a product of its factors. For example, turning a sum or difference of terms into a product of simpler expressions that, when multiplied, will give back the original expression.
In the given problem, we used the difference of squares identity to factor the expression \((a+b)^2 - (a-b)^2\). First, identify the values for \(x\) and \(y\) corresponding to the identity: \(x = (a+b)\) and \(y = (a-b)\). Then apply the identity to get:
  • \((a+b) + (a-b)\)
  • \((a+b) - (a-b)\)
These determined factors are easier to manage and simplify further. Factoring helps to solve equations, simplify expressions, and understand math better.
Simplifying Algebraic Expressions
Simplifying algebraic expressions removes redundancy and presents expressions in their simplest form. It makes it easier to perform calculations and solve problems.
Once we factored the expression \((a+b)^2-(a-b)^2\) into \([(a+b)+(a-b)][(a+b)-(a-b)]\), the next task was simplifying each factor. Simplification involved basic operations:
  • \((a+b) + (a-b)\) simplifies to \(2a\)
  • \((a+b) - (a-b)\) simplifies to \(2b\)
This straightforward simplification turned the expression into \((2a)(2b)\), which further combines to \(4ab\). This process of reducing factors to their simplest form is essential for effective problem-solving.

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Most popular questions from this chapter

Verify these formulas by expanding and simplifying the right-hand side. $$\begin{array}{l}A^{2}-1=(A-1)(A+1) \\\A^{3}-1=(A-1)\left(A^{2}+A+1\right) \\\A^{4}-1=(A-1)\left(A^{3}+A^{2}+A+1\right)\end{array}$$ Based on the pattern displayed in this list, how do you think \(A^{5}-1\) would factor? Verify your conjecture. Now generalize the pattern you have observed to obtain a factoring formula for \(A^{n}-1,\) where \(n\) is a positive integer.

Factor the expression completely. $$49-4 y^{2}$$

Sketch the region given by the set. $$\left\\{(x, y) | x^{2}+y^{2} \leq 1\right\\}$$

Factor the expression completely. $$y^{4}(y+2)^{3}+y^{5}(y+2)^{4}$$

A square field in a certain state park is mowed around the edges every week. The rest of the field is kept unmowed to serve as a habitat for birds and small animals (see the figure). The field measures \(b\) feet by \(b\) feet, and the mowed strip is \(x\) feet wide. (a) Explain why the area of the mowed portion is \(b^{2}-(b-2 x)^{2}\) (b) Factor the expression in (a) to show that the area of the mowed portion is also \(4 x(b-x)\) CAN'T COPY THE GRAPH
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