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Chapter 6: Problem 15

Factor the given expressions completely. $$2 x+4 y-8 z$$

Short Answer

Expert verified
The expression factors to \(2(x + 2y - 4z)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, look at each term in the expression \(2x + 4y - 8z\). The coefficients are 2, 4, and -8. Determine the greatest common factor of these numbers. The GCF of 2, 4, and 8 is 2.
02

Factor Out the GCF

Once you have identified the GCF as 2, factor it out from the entire expression. Divide each term by the GCF: \[2x + 4y - 8z = 2(x) + 2(2y) - 2(4z).\] Therefore, the expression with the GCF factored out is \[2(x + 2y - 4z).\]

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Key Concepts

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

Greatest Common Factor
When you're faced with the task of factoring an expression like \(2x + 4y - 8z\), finding the Greatest Common Factor (GCF) is a crucial first step.
The GCF is the largest number that evenly divides all the numbers in the given expression. This means you should look at the coefficients in the terms. These are the numbers directly in front of the variables (in this case, 2, 4, and -8).
  • To determine the GCF, find the highest number that can divide each coefficient, without leaving a remainder.
  • For our exercise, 2, 4, and -8 can all be divided by 2 not only without a remainder but because 2 is indeed the greatest number capable of this task.
  • Hence, the GCF of 2, 4, and -8 is 2.
Finding the GCF simplifies the next steps of factorization, helping to reduce equations to their simplest form. This paves the way for the smooth tackling of algebraic expressions.
Step by Step Factorization
Once the Greatest Common Factor is identified, you can proceed with what is called step by step factorization. This process simplifies the expression by extracting the GCF from each term, making it easier to understand or solve. It involves breaking down the original equation into a simpler form. Here's how you do it:
  • First, take the GCF, which we've determined to be 2, and use it to divide each term in the expression. This ensures that the GCF is effectively 'factored out'.
  • In our example, the expression \(2x + 4y - 8z\) transforms into \(2(x + 2y - 4z)\).
  • The process shows how factoring makes the whole expression simpler by grouping terms.
Throughout factorization, you are rewriting the expression in terms of the common multipliers, providing an elegant and user-friendly way to handle complex mathematical statements.
Mathematical Expressions
Understanding factoring hinges on grasping the basics of mathematical expressions. These expressions are a combination of numbers, variables, and operations like addition or subtraction. They are the heart of algebra and other branches of mathematics.
Every portion of an expression, like \(2x + 4y - 8z\), has a purpose. Variables (letters) represent numbers that can vary.
  • The constant numbers in front of the variable letters, known as coefficients, can often be simplified by identifying and factoring out the greatest common factor.
  • Breaking down expressions into their simplest form helps to resolve equations, find solutions, and understand the relationships between terms.
  • It鈥檚 key to recognize that each mathematical expression needs to be correctly managed and simplified whenever possible before solving.
As you develop your skills with these expressions through factoring, you build a foundation for tackling more advanced math topics, preparing you to solve increasingly complex challenges.

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Most popular questions from this chapter

Solve the given problems. Find three values of \(k>0\) such that \(x^{2}-6 x+k\) can be factored.

Factor the given expressions completely. Each is from the technical area indicated. $$3 Q^{2}+Q-30 \quad \text { (fire science) }$$

In Exercises \(71-76,\) reduce each fraction to simplest form. Each is from the indicated area of application. $$\frac{m u^{2}-m v^{2}}{m u-m v} \quad \text { (nuclear energy) }$$

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Solve the given problems. Is it true that \((1-2 x)^{3}=(2 x-1)^{2}(1-2 x) ?\)
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