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Chapter 0: Problem 140

Factor completely. $$x^{4}-y^{4}-2 x^{3} y+2 x y^{3}$$

Short Answer

Expert verified
The completely factored form of the polynomial \(x^{4}-y^{4}-2x^{3}y+2xy^{3}\) is \((x-y)(x+y)(x^{2}+y^{2} - 2xy)\).

Step by step solution

01

Identify the Difference of Squares

The first two terms \(x^{4} - y^{4}\) can be identified as a difference of squares. A difference of squares can be factored following the formula \(a^{2} - b^{2} = (a-b)(a+b)\). Applying this to the first two terms results in \((x^{2}-y^{2})(x^{2}+y^{2})\).
02

Factor Out Common Terms

The last two terms \(-2x^{3}y + 2xy^{3}\) can be factored by taking out the common factor, which is \(2xy(x^2 - y^2)\).
03

Combine Both Parts

Both parts from steps 1 and 2 can be combined as follows: \(x^{4}-y^{4}-2x^{3}y+2xy^{3} = (x^{2}-y^{2})(x^{2}+y^{2}) - 2xy(x^{2}-y^{2})\).
04

Factor Out Common Term

At this point, it is noticed that \((x^{2}-y^{2})\) is a common term in both terms of the expression. Therefore, the expression can be factored out to give \((x^{2}-y^{2})(x^{2}+y^{2} - 2xy)\).
05

Factor the Difference of Squares Further

The term \((x^{2}-y^{2})\) is a difference of squares and can be factored further into \((x-y)(x+y)\). Therefore, the fully factored form of the given polynomial is \((x-y)(x+y)(x^{2}+y^{2} - 2xy)\).

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Most popular questions from this chapter

Explain how to determine which numbers must be excluded from the domain of a rational expression.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$6+\frac{1}{x}=\frac{7}{x}$$

What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) $$\begin{aligned} &2>1\\\ &2(y-x)>1(y-x)\\\ &2 y-2 x>y-x\\\ &\begin{aligned} y-2 x &>-x \\ y &>x \end{aligned} \end{aligned}$$ The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\)

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