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Chapter 5: Problem 66

Find each product. $$(x+y)\left(x^{2}+5 x y+y^{2}\right)$$

Short Answer

Expert verified
\(x^3+5x^2y+xy^2+yx^2+5y^2x+y^3\)

Step by step solution

01

Separate the Terms

Divide the problem into smaller parts by separating the terms. Here we have \(x\) and \(y\) in the binomial, and \(x^2\), \(5xy\), and \(y^2\) in the trinomial.
02

Distribute the Terms

The multiplication of each term in the binomial (x + y) with each term in the trinomial ( \(x^2+5xy+y^2\) ) is performed. This reflects the distributive property of multiplication over addition.
03

Perform the Multiplication

Perform the multiplications: \(x * x^2\), \(x * 5xy\), \(x * y^2\), \(y * x^2\), \(y * 5xy\), and \(y * y^2\). The resulting terms are \(x^3\), \(5x^2y\), \(xy^2\), \(yx^2\), \(5y^2x\), and \(y^3\).
04

Combine Like Terms

The terms \(x * 5xy\) and \(y * 5xy\) (which are \(5x^2y\) and \(5y^2x\) respectively) are similar and are combined to get the resulting expression.

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