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Chapter 5: Problem 45

Multiply. $$ (3 z-u)(2 z+3 u) $$

Short Answer

Expert verified
6z^2 + 7zu - 3u^2

Step by step solution

01

Apply the Distributive Property (FOIL Method)

To multiply \((3z - u)(2z + 3u)\), use the distributive property, also known as the FOIL (First, Outer, Inner, Last) method.
02

Multiply the First Terms

Multiply the first terms in each binomial: \((3z) \cdot (2z) = 6z^2\).
03

Multiply the Outer Terms

Multiply the outer terms in each binomial: \((3z) \cdot (3u) = 9zu\).
04

Multiply the Inner Terms

Multiply the inner terms in each binomial: \((-u) \cdot (2z) = -2uz\).
05

Multiply the Last Terms

Multiply the last terms in each binomial: \((-u) \cdot (3u) = -3u^2\).
06

Sum All Products

Add all the products obtained: \[6z^2 + 9zu - 2uz - 3u^2\]
07

Combine Like Terms

Combine any like terms from the expression: \[6z^2 + 7zu - 3u^2\]

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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 principle in algebra, often used to simplify expressions and solve equations. When using the distributive property, you multiply a single term by each term within a parenthesis. For example, if we have a binomial \((a+b)(c+d)\), we distribute each term in the first parenthesis to each term in the second one:
\[ (a+b)(c+d) \] becomes
\[a \cdot c + a \cdot d + b \cdot c + b \cdot d.\]

This helps in breaking down complex expressions into simpler parts that are easier to work with. In our exercise, the distributive property is applied through the FOIL method, which specifically deals with binomials.
Binomial Multiplication
Binomial multiplication is a specific application of the distributive property where two binomials are multiplied together. The FOIL method is a handy mnemonic to remember this process:
  • **First:** Multiply the first terms in each binomial.
  • **Outer:** Multiply the outer terms in the binomials.
  • **Inner:** Multiply the inner terms in the binomials.
  • **Last:** Multiply the last terms in each binomial.

For example, in our exercise \((3z - u)(2z + 3u)\):
\

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Most popular questions from this chapter

Write equations for two functions \(f\) and \(g\) such that the domain of \(f+g\) is \(\\{x | x \text { is a real number and } x \neq-2 \text { and } x \neq 5\\}\)

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