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Chapter 5: Problem 68

Find each product. $$(x-y)\left(x^{2}-4 x y+y^{2}\right)$$

Short Answer

Expert verified
The short answer to the question is: \[ x^{3} - 5x^{2}y + 4xy^{2} + y^{3} \]

Step by step solution

01

Distribute \(x - y\) to each term.

The binomial \(x - y\) must be distributed to each term in the trinomial. That means multiplying \(x - y\) by \(x^{2}\), \(-4xy\), and \(y^{2}\). \[ (x - y) x^{2} , (x - y)(-4xy), (x - y)y^{2} \] respectively.
02

Multiply each pair of terms.

Following the rule of multiplication, the product of \(x - y\) and each term in the trinomial is as follows: \[ x^{3} - yx^{2}, -4x^{2}y + 4xy^{2}, y^{3} - y^{2}y \] respectively.
03

Simplify the Equation.

The simplified form of our previous step is: \[ x^{3} - yx^{2} -4x^{2}y + 4xy^{2} + y^{3} - y^{2}y \] Now combine like terms to get the final result: \[ x^{3} - yx^{2} -4x^{2}y + 4xy^{2} + y^{3} - y^{3}\]

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