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Chapter 5: Problem 126

Factor out the specified factor. $$ y^{n} \text { from } 2 y^{n+2}-3 y^{n+3} $$

Short Answer

Expert verified
The expression is factored as \(y^{n}(2y^{2} - 3y^{3})\).

Step by step solution

01

Understand the Problem

The problem asks us to factor out the term \(y^{n}\) from the expression \(2y^{n+2} - 3y^{n+3}\). This means we need to rewrite the expression in a way that separates \(y^{n}\) from the rest of the terms.
02

Factor the Term from Each Part

We look at each term in the expression individually. For \(2y^{n+2}\), we can write it as \(2y^{n}y^{2}\) because \(y^{n+2} = y^{n}y^{2}\). Similarly, for \(-3y^{n+3}\), we can write it as \(-3y^{n}y^{3}\) because \(y^{n+3} = y^{n}y^{3}\).
03

Pull Out the Common Factor

Now that we have expressed both terms with \(y^{n}\) as a factor, we can factor \(y^{n}\) out of the entire expression: \(y^{n}(2y^{2} - 3y^{3})\).
04

Rewrite the Expression

After factoring \(y^{n}\) out, our expression becomes \(y^{n}(2y^{2} - 3y^{3})\), which is the factored form of the original expression.

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are foundational components in mathematics, especially in algebra. They consist of variables, numbers, and arithmetic operations. In the expression \(2y^{n+2} - 3y^{n+3}\), each part separated by a '+' or '-' sign is known as a term. Characteristics of an Algebraic Expression:
  • It can include constants (fixed numbers), variables (like \(y\)), and coefficients (like 2 and -3).
  • Operations such as addition, subtraction, multiplication, and division are used.
In the given problem, the algebraic expression is translated into a form that factors out one of the variables. Understanding how each component works is crucial for breaking down and simplifying complex expressions.
Common Factor
A common factor in algebra is something shared between two or more terms. It is the component we can "factor out" from an expression, simplifying or rewriting it in another form. How to Identify a Common Factor:
  • Look for the greatest expression that evenly divides all terms in the algebraic expression.
  • In our expression \(2y^{n+2} - 3y^{n+3}\), the common factor was \(y^n\) because both terms include it.
  • By factoring \(y^n\), we simplify the original expression to \(y^n(2y^{2} - 3y^{3})\).
Recognizing common factors allows you to rewrite expressions more simply and makes solving equations easier. This skill is invaluable in algebra and beyond.
Exponents and Powers
Exponents are a shorthand way to denote repeated multiplication of a number by itself. In an expression like \(y^n\), \(n\) is the exponent, and \(y\) is the base. Basic Rules of Exponents:
  • When multiplying terms with the same base, add their exponents: \(y^a \cdot y^b = y^{a+b}\).
  • In our problem, this rule helps to rewrite terms as \(2y^{n}y^{2}\) and \(-3y^{n}y^{3}\).
  • The power of a power rule also states \((y^a)^b = y^{a \cdot b}\).
By applying these exponent rules, one can systematically break down and factor expressions. Understanding exponents is key to managing algebraic expressions efficiently.

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Most popular questions from this chapter

Factor. Assume all variables represent natural numbers. $$ a^{3 b}-c^{3 b} $$

Factor each expression. $$ y^{2}-(2 x-t)^{2} $$

On a test, when asked to find \((x-y)^{2},\) a student answered \(x^{2}-y^{2} .\) What error did the student make?

Look Alikes \(. .\) \(*\) $$ \text { a. } 3 y+\left(y^{7}+y\right)+\left(y^{3}+y\right) \text { b. } 3 y\left(y^{7}+y\right)\left(y^{3}+y\right) $$

Factor. Assume all variables represent natural numbers. $$ 27 x^{3 n}+y^{3 n} $$
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