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Chapter 5: Problem 72

Factor completely. \(100-x^{2}-2 x y-y^{2}\)

Short Answer

Expert verified
The expression \( 100 - x^2 - 2xy - y^2 \) factors completely to \( (10 - x - y)(10 + x + y) \).

Step by step solution

01

Rearrange the Terms

First, rewrite the expression in a recognizable form. The given expression is: \[ 100 - x^{2} - 2xy - y^{2} \]Rearrange it to write similar terms together:\[ 100 - (x^{2} + 2xy + y^{2}) \]
02

Recognize the Pattern

Notice that the expression inside the parentheses is a perfect square trinomial:\[ x^{2} + 2xy + y^{2} = (x + y)^{2} \]So, rewrite the expression as:\[ 100 - (x + y)^{2} \]
03

Apply Difference of Squares

The expression is now in the form of a difference of squares: \[ a^{2} - b^{2} = (a - b)(a + b) \]Here, let \( a = 10 \) (since \( 100 = 10^{2} \)) and \( b = (x + y) \), thus:\[ 100 - (x + y)^{2} = 10^{2} - (x + y)^{2} \]This can be factored as:\[ (10 - (x + y))(10 + (x + y)) \]
04

Simplify the Factors

Distribute and combine like terms inside the parentheses:\[ (10 - x - y)(10 + x + y) \]

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Key Concepts

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

Perfect Square Trinomials
A perfect square trinomial is an algebraic expression that follows the form:
  • \(a^2 + 2ab + b^2 = (a+b)^2\) or
  • \(a^2 - 2ab + b^2 = (a-b)^2\).

In our exercise, the term inside the parentheses, \(x^2 + 2xy + y^2\), fits the first form.
It can be rewritten as \((x+y)^2\). Recognizing these patterns simplifies the factoring process.
It helps us transform complex expressions into more manageable forms.
Difference of Squares
The difference of squares is another important algebraic identity. It states that any expression of the form \[a^2 - b^2\] can be factored into \((a - b)(a + b)\).
In our exercise, after recognizing the perfect square trinomial, we rewrite it to match this identity: \[100 - (x + y)^2\].
This can be seen as \((10)^2 - (x + y)^2\), where \(a = 10\) and \(b = (x + y)\). Using the difference of squares identity, we factored it into \((10 - (x + y))(10 + (x + y))\).
Recognizing and applying these patterns allows for efficient factoring of expressions.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions using various algebraic techniques.
In our exercise, the initial step involved recognizing how to rearrange the terms: from \(100 - x^2 - 2xy - y^2\) to \(100 - (x^2 + 2xy + y^2)\).
Rearranging helped us recognize the perfect square trinomial. Breaking down complex expressions into familiar parts uses algebraic manipulation.
It simplifies solving and factoring algebraic problems.
Binomials
A binomial is an algebraic expression with two terms separated by addition or subtraction. In our final factored form: \((10 - x - y)(10 + x + y)\), we have two binomials.
Each binomial represents a part of the difference of squares' factors.
Understanding binomials and manipulating them is key to solving algebraic expressions. It aids in breaking down and simplifying complex problems. Binomials are fundamental in various algebraic identities and operations.

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