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Chapter 5: Problem 26

Find each product. When possible, write down only the answer. $$(z-1)(y+2)$$

Short Answer

Expert verified
zy + 2z - y - 2

Step by step solution

01

Distribute the terms

Multiply each term in the first parenthesis by each term in the second parenthesis. This means distributing \(z\) and \(-1\) to \(y \) and \(+2\).
02

Multiply the first terms

Multiply \(z\) by \(y\), which gives \(zy\).
03

Multiply the outer terms

Multiply \(z\) by \(+2\), which gives \(+2z\).
04

Multiply the inner terms

Multiply \(-1\) by \(y\), which gives \(-y\).
05

Multiply the last terms

Multiply \(-1\) by \(+2\), which gives \(-2\).
06

Combine the terms

Combine all the terms from steps 2, 3, 4, and 5: \(zy + 2z - y - 2\). This is the expanded form of the binomials.

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Key Concepts

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

Polynomial Operations
Polynomial operations involve various arithmetic actions such as addition, subtraction, multiplication, and division on polynomials. When we talk about polynomials, we are referring to expressions consisting of terms that contain variables raised to different powers. Examples range from simple expressions like (z + 2) to more complex ones like (z^2 - 3z + 4).
Multiplying polynomials is one of the key operations, and it requires us to understand some specific steps and properties.
In the exercise given, we deal with a binomial multiplication (z-1)(y+2).
Our task is to expand and simplify this expression
Distributive Property
The distributive property is a fundamental rule in algebra. It allows you to distribute or spread out multiplication over addition or subtraction within an expression. Simply put, it means: \[a(b + c) = ab + ac\].
In our exercise, we need to use this property to multiply two binomials: (z-1) and (y+2).
This means every term in the first binomial needs to be multiplied by every term in the second binomial. Let's break it down:
  • Multiply (z) by (y), which gives us (zy).
  • Multiply (z) by (+2), resulting in (+2z).
  • Multiply (-1) by (y), giving (-y).
  • Finally, multiply (-1) by (+2), leading to (-2).
So, the multiplication using distributive property would result in four terms: (zy + 2z - y - 2)
Combining Like Terms
Once you've used the distributive property in polynomial operations, you often end up with multiple terms, some of which may be similar. Combining like terms simplifies the expression.
Like terms are terms that have the same variable raised to the same power. For example, (2z) and (3z) are like terms, while (2z) and (2y) are not.
In our exercise, after distributing, we get the terms: (zy + 2z - y - 2).
Since there are no like terms to combine in this example, the expression remains: (zy + 2z - y - 2).
This is our final simplified form

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