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Chapter 4: Problem 30

Factor out the GCF. \(y^{2}(y-1)+9(y-1)\)

Short Answer

Expert verified
The GCF is \((y-1)\), so the expression factors to \((y-1)(y^{2} + 9)\).

Step by step solution

01

Identify Common Factor

Observe the expression is composed of two terms: \(y^{2}(y-1)\) and \(9(y-1)\). Both terms have a common factor of \((y-1)\).
02

Rewrite Expression with Common Factor

Express each term as a product of the identified common factor: \( (y-1)(y^{2} + 9) \). The GCF here is \((y-1)\).
03

Verify Factoring

Re-distribute \((y-1)\) back into \(y^{2} + 9\) to ensure that the expression returns to its original form, \( y^{2}(y-1) + 9(y-1) \). This confirms the factoring is correct.

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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 algebraic concept used in simplifying expressions. In simple terms, it is the largest factor that divides two or more numbers or terms. Finding the GCF is an essential step in factoring polynomials. It allows you to simplify expressions and make them easier to work with. How to Find the GCF?
  • List Factors: Find all the factors for each of your terms.
  • Determine Commonality: Identify which factors are common among the terms involved.
  • Select the Greatest: Pick the largest factor that appears in all term factor lists.
In the given exercise: Both terms in the expression, \(y^2(y-1)\) and \(9(y-1)\), share a common factor of \((y-1)\). Thus, \((y-1)\) is their greatest common factor.
Algebraic Expressions
An **algebraic expression** consists of numbers, variables, and arithmetic operations (like addition, subtraction, etc.). They represent a specific calculation or set of calculations to be performed. Algebraic expressions can vary from simple to quite complex, consisting of many terms. Components of Algebraic Expressions:
  • Variables: Symbols like \(x, y, z\) that represent numbers.
  • Coefficients: Numbers that multiply the variables.
  • Constants: Standalone numbers.
  • Operators: Signs that represent operations like + or -.
The exercise involves the expression \(y^2(y-1) + 9(y-1)\), where each part is an algebraic expression. The challenge is to factor it using the GCF, aiding in simplifying the problem.
Polynomial Expressions
**Polynomial expressions** are a type of algebraic expression that includes sums of powers of variables, typically modified by coefficients. They form an essential part of algebra and calculus.Structure of Polynomial Expressions:
  • Terms: Each separate part of a polynomial, like \(3x^2\), \(-5x\), or \(+7\).
  • Degree: The highest power of the variable (e.g. in \(x^3 - 4x + 8\), the degree is 3).
  • Coefficients: Numbers in front of the variables (e.g. in \(4x^2\), 4 is the coefficient).
In the exercise, both \(y^2(y-1)\) and \(9(y-1)\) are polynomial expressions, where terms have been combined through operations. The task is to factor the given polynomial by finding the common factor and simplifying the expression into \((y-1)(y^2 + 9)\).

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