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Chapter 0: Problem 59

Find each product. $$(x+5 y)(7 x+3 y)$$

Short Answer

Expert verified
The product of the binomials \((x+5y)(7x+3y)\) is \(7x^2 + 38xy + 15y^2\).

Step by step solution

01

Distribute First Term of the First Binomial

The first step is to distribute the first term of the first binomial, \(x\), to each term of the second binomial, \(7x + 3y\). This yields:\(x * 7x + x * 3y\), which simplifies to \(7x^2 + 3xy\)
02

Distribute Second Term of the First Binomial

The second step involves distributing the second term of the first binomial, \(5y\), to each term of the second binomial. This looks like: \(5y * 7x + 5y * 3y\), which simplifies to \(35xy + 15y^2\)
03

Combine Like Terms

Now, combine like terms for the final answer. There are no like terms here, so the final result is the sum of the products from Steps 1 and 2: \(7x^2 + 3xy + 35xy + 15y^2\)
04

Combine Coefficients of \(xy\)

The terms \(3xy\) and \(35xy\) can be combined as they have similar variable parts, yielding the final answer. The sum of coefficients is \(38\) hence the result becomes \(7x^2 + 38xy + 15y^2\)

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