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Chapter 6: Problem 60

Write in factored form by factoring out the greatest common factor. \(y^{2}(x-9)+1(x-9)\)

Short Answer

Expert verified
\( (x-9)(y^{2} + 1) \)

Step by step solution

01

Identify the common factor

Look at each term in the expression and identify the common factor. In this case, the common factor is \( (x-9) \).
02

Factor out the common factor

Factor out the greatest common factor \( (x-9) \) from each term: \( (x-9)(y^{2} + 1) \).

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Key Concepts

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

Greatest Common Factor
The greatest common factor (GCF) is a key concept in algebra, especially when factoring expressions. The GCF is the largest factor that is common to all terms in an expression. For example, in the expression given in the exercise, \( y^{2}(x-9) + 1(x-9) \), the common factor is \( (x-9) \).
To find the GCF, examine each term separately and identify the elements that are the same in every term. In this exercise, both terms include \( (x-9) \), so it is the greatest common factor. Once the GCF is identified, you can factor it out of the expression to simplify it.
Factored Form
Factoring is the process of breaking down an expression into a product of simpler expressions. When you 'factor out' the GCF, you're simplifying the original expression. In the exercise, the original expression \( y^{2}(x-9) + 1(x-9) \) was simplified to \( (x-9)(y^{2} + 1) \).
Here's how it works:
  • First, identify the GCF, which is \( (x-9) \).
  • Then, rewrite the original expression as a product of the GCF and another expression. This means you take \( (x-9) \) out of both terms: \( (x-9)(y^{2} + 1) \).
This is your final factored form! Factoring is essential for solving equations and simplifying expressions in algebra.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operation symbols. Expressions can include terms that are added, subtracted, multiplied, or divided. For instance, in our exercise, the expression \( y^{2}(x-9) + 1(x-9) \) includes variable terms \( y^{2} \) and \( x-9 \), and a constant term with \( 1 \).
When working with algebraic expressions, it's important to:
  • Identify terms: Separate parts of the expression that are added or subtracted.
  • Simplify where possible: Combine like terms and factor out common factors.
Practicing these steps helps in understanding more complex algebraic problems and in solving them efficiently.

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