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Chapter 5: Problem 60

The following exercises are of mixed variety. Factor each polynomial. $$ x^{2}-2 x y+y^{2}-4 $$

Short Answer

Expert verified
(x-y-2)(x-y+2)

Step by step solution

01

- Recognize the Structure

Observe that the given polynomial is a quadratic expression and can be rearranged to a recognizable form. The polynomial is: \[ x^{2} - 2xy + y^{2} - 4 \]
02

- Group Terms

Group the first three terms as part of a perfect square and the constant term separately: \[ (x^{2} - 2xy + y^{2}) - 4 \]
03

- Identify the Perfect Square

Recognize that \(x^{2} - 2xy + y^{2}\) is a perfect square trinomial and can be written as: \[ (x-y)^{2} \]
04

- Substitute Back

Replace \(x^{2} - 2xy + y^{2}\) with \((x-y)^{2}\): \[ (x-y)^{2} - 4 \]
05

- Apply Difference of Squares

Notice that \((x-y)^{2} - 4\) is a difference of squares. It can be factored further into: \[ ((x-y) - 2)((x-y) + 2) \]This simplifies to: \[ (x-y-2)(x-y+2) \]

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Key Concepts

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

quadratic expressions
Quadratic expressions are a type of polynomial characterized by their highest power being 2. A standard quadratic expression looks like this:

\( ax^{2} + bx + c \) The terms involve a variable (often x) raised to the second power, a term with the variable raised to the first power, and a constant term. Recognizing the structure of a quadratic expression is key to factoring it correctly.

In our problem, \(x^{2} - 2xy + y^{2} - 4\), we identified that it contains a quadratic part \(x^{2} - 2xy + y^{2}\). By focusing on this part, we can apply special factoring techniques to simplify further.
perfect square trinomial
A perfect square trinomial is a specific type of quadratic expression. It takes the form:

\( a^{2} 卤 2ab + b^{2} \) These trinomials can be factored into:

\( (a 卤 b)^{2} \) Recognizing these patterns saves time and simplifies the factoring process.

In our case, we found that: \(x^{2} - 2xy + y^{2}\) fits the form of a perfect square trinomial and can be rewritten as: \( (x-y)^{2} \) This was a vital step in simplifying our factoring process.
difference of squares
The difference of squares is another valuable factoring technique. It applies when you have two terms, each of which is a perfect square, separated by a subtraction sign.

The general form looks like: \( a^{2} - b^{2} \),

and it can be factored into: \( (a - b)(a + b) \).

This method simplifies factoring and solving polynomial expressions considerably.

In our problem, we identified that: \( (x-y)^{2} - 4 \) is a difference of squares. Applying the formula: it becomes \( ((x-y) - 2)((x-y) + 2) \). The problem, therefore, simplifies to \( (x-y-2)(x-y+2) \).

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Most popular questions from this chapter

The following exercises are of mixed variety. Factor each polynomial. $$ z^{4}-9 z^{2}+20 $$

Solve each equation for the specified variable. $$ 4 s+7 p=t p-7 \text { for } p $$

Solve each equation. $$ x^{3}-6 x^{2}=-8 x $$

If a rock is dropped from a building 576 ft high, then its distance in feet from the ground \(t\) seconds later is modeled by the function $$ f(t)=-16 t^{2}+576 $$ How long after it is dropped will it hit the ground?

Solve each equation for the specified variable. $$ c=\frac{-2 t+4}{t} \text { for } t $$
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