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Chapter 0: Problem 69

In Exercises 67鈥�82, find each product. $$(x-3 y)(2 x+7 y)$$

Short Answer

Expert verified
The product of \((x - 3y)\) and \((2x + 7y)\) is \(2x^2 + xy - 21y^2\).

Step by step solution

01

Distribute \(x\) from first binomial

Initially, distribute the \(x\) to every term in the second binomial: \(x \times 2x\) and \(x \times 7y\), which results in \(2x^2 + 7xy\)
02

Distribute \(-3y\) from the first binomial

Then, distribute \(-3y\) to each term in the second binomial: \(-3y \times 2x\) and \(-3y \times 7y\), which results in \(-6xy - 21y^2\)
03

Add the results

Combine the results of Step 1 and Step 2: \(2x^2 + 7xy - 6xy - 21y^2\)
04

Simplify

Combine like terms: \(2x^2 + xy - 21y^2\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental algebraic tool that helps in multiplying expressions like polynomials. It allows you to take a single term outside a parenthesis and multiply it by each term inside the parenthesis. In simpler terms, when you have a multiplication like \((a + b)(c + d)\), you distribute each term from the first binomial across all terms in the second binomial.
In our example, \((x-3y)(2x+7y)\), the distributive property is used twice:
  • First, distribute \(x\) from \(x - 3y\) across \(2x + 7y\). This involves calculating \(x \times 2x = 2x^2\) and \(x \times 7y = 7xy\).
  • Second, distribute \(-3y\) in the same way: \(-3y \times 2x = -6xy\) and \(-3y \times 7y = -21y^2\).
By using the distributive property, you break down complex polynomials into manageable parts. Such step is essential in transforming polynomials from complex expressions to simplified forms.
Binomials
A binomial is a polynomial with exactly two terms, which are typically separated by either a plus or a minus sign. For example, expressions like \(x - 3y\) and \(2x + 7y\) are both binomials. Understanding binomials is crucial when applying operations such as multiplication.
To multiply two binomials, each term in the first must multiply each term in the second. This ensures that all possible products are accounted for. The typical strategy involves using the distributive property to simplify multiplication:
  • First Binomial: Take each term and multiply it separately by each term in the second binomial.
  • Resulting Terms: In our example, \(x \times 2x\), \(x \times 7y\), \(-3y \times 2x\), and \(-3y \times 7y\) produce all terms of the result.
By mastering how to operate binomials, one simplifies polynomial multiplication and gains deeper insight into algebraic expressions.
Combining Like Terms
Combining like terms is the process of simplifying polynomials by merging terms that have the same variable parts into a single term. This is crucial after multiplying polynomials to ensure the expression is as simple as possible.
In the context of our problem, after distributing terms, you'll have:
  • \(2x^2\), a term on its own as no other term shares the same variables.
  • \(7xy\) and \(-6xy\), where the like terms can be combined since they both have the same variable product \(xy\).
  • \(-21y^2\), also distinct, remaining as is.
Combining \(7xy - 6xy\) results in \(1xy\) (or \(xy\)).
Thus, the expression simplifies to \(2x^2 + xy - 21y^2\). By efficiently combining like terms, you reduce clutter in expressions, leading to easily interpretable results.

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