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Chapter 7: Problem 190

Factor. \(12 x^{2}+36 y-24 z\)

Short Answer

Expert verified
12(x^2 + 3y - 2z)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, identify the greatest common factor (GCF) of the terms in the expression. The terms are 12, 36, and 24. The GCF of these numbers is 12.
02

Factor out the GCF

Next, factor out the GCF from each term in the expression. The expression becomes: \[ 12(x^2) + 12(3y) - 12(2z) \]This can be rewritten as: \[ 12(x^2 + 3y - 2z) \]
03

Verify the factorization

Finally, verify the factorization by distributing the 12 back into the parentheses to check if it equals the original expression: \[ 12(x^2 + 3y - 2z) = 12x^2 + 36y - 24z \]Since the expression matches the original, the factorization 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
Understanding the Greatest Common Factor (GCF) is crucial when working on polynomial factorization. The GCF is the highest number that can divide all the terms in a polynomial without leaving a remainder. For example, if we have the numbers 12, 36, and 24, the largest number that divides each of these perfectly is 12. Here's a simple way to find the GCF:
  • List the factors of each number.
  • Identify the common factors.
  • Choose the greatest one.
In our exercise, 12 is the GCF of the polynomial terms 12, 36, and 24. Recognizing this is key for simplifying and factoring polynomials.
Polynomial Factorization
Polynomial factorization involves breaking down an expression into simpler components, or 'factors', that, when multiplied together, give the original expression. After identifying the GCF, we can divide each term of the polynomial by the GCF to factor it out.
In the given exercise, we factored out the GCF (12) from each term in the polynomial:
\( 12x^2 + 36y - 24z = 12(x^2 + 3y - 2z) \)
To check our work, we use the distributive property to multiply back the GCF into the simplified polynomial, confirming it matches the original polynomial:
\( 12(x^2 + 3y - 2z) = 12x^2 + 36y - 24z \)
Factoring polynomials simplifies expressions and is crucial for solving algebraic equations.
Algebraic Expressions
Algebraic expressions contain numbers, variables, and operations like addition, subtraction, multiplication, and division. They can represent real-world situations, making it easier to analyze and solve problems.
Consider the given algebraic expression:
\( 12x^2 + 36y - 24z \)
This polynomial has three terms. Factoring helps simplify such expressions and makes solving equations more manageable.
The key steps are:
  • Identify the GCF of the coefficients.
  • Factor out the GCF.
  • Verify by redistributing to ensure you return to the original expression.
Understanding and manipulating algebraic expressions is fundamental in algebra and beyond!

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