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Chapter 0: Problem 70

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

Short Answer

Expert verified
Therefore, the product of \((x+y)\) and \(\left(x^{2}-x y+y^{2}\right)\) is \(x^{3} - x^{2}y + x y^{2} + y x^{2} - y^{2}x + y^{3}\).

Step by step solution

01

Expand the first term

Start by expanding the first term. Multiply \(x\) from \(x+y\) with each term in \(\left(x^{2}-x y+y^{2}\right)\). This yields following three terms: \(x . x^{2} = x^{3}\), \(x . (- xy) = - x^{2} y\), \(x . y^{2} = x y^{2}\). The partial result is then \(x^{3} - x^{2}y + x y^{2}\).
02

Expand the second term

Next, expand the second term. Multiply \(y\) from \(x+y\) with each term in \(\left(x^{2}-x y+y^{2}\right)\). This yields following three terms: \(y . x^{2} = y x^{2}\), \(y . (- xy) = - y^{2} x\), \(y . y^{2} = y^{3}\). The partial result is then \(y x^{2} - y^{2}x + y^{3}\).
03

Add the results

Now, add the results from Step 1 and Step 2 to generate the final result. Thus the answer becomes \(x^{3} - x^{2}y + x y^{2} + y x^{2} - y^{2}x + y^{3}\).

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