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Chapter 1: Problem 155

Find an equivalent expression by factoring. $$ 7 x-21 y+14 z $$

Short Answer

Expert verified
7(x - 3y + 2z)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

The first step is to determine the greatest common factor for all the terms in the expression. The terms are: 7x, -21y, and 14z. The GCF of 7, -21, and 14 is 7.
02

Factor Out the GCF

Divide each term by the GCF (7) and write the expression as a product of the GCF and the resulting terms within parentheses. So, \[7x - 21y + 14z = 7(x - 3y + 2z)\]

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Key Concepts

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

Greatest Common Factor
To factor an algebraic expression, we first need to identify the greatest common factor (GCF) in each term of the expression. The GCF is the highest number that divides evenly into each of the individual numbers or terms.

For the expression given in the exercise, the terms are 7x, -21y, and 14z. We look for the largest number that completely divides 7, -21, and 14 without leaving a remainder.
  • 7 divides 7 evenly (7 梅 7 = 1)
  • 7 divides -21 evenly (-21 梅 7 = -3)
  • 7 divides 14 evenly (14 梅 7 = 2)
This means our GCF is 7. Recognizing the GCF simplifies the factoring process, which is crucial to creating equivalent but simpler expressions.
Factoring
Once we have identified the greatest common factor (GCF) as 7, the next step is to factor it out of the entire expression. Factoring involves rewriting each term in the expression as a product of the GCF and another term.

By dividing each term in the expression by the GCF, we get:
  • 7x 梅 7 = x
  • -21y 梅 7 = -3y
  • 14z 梅 7 = 2z
This allows us to re-write the original expression, 7x - 21y + 14z, as a product of the GCF and the simplified terms within parentheses: \[ 7(x - 3y + 2z) \].
This is the factored form, where the complex expression is now much easier to work with.
Algebraic Expressions
Algebraic expressions involve variables, constants, and operations (like addition, subtraction). They can represent real-world problems and mathematical relationships.
In the exercise, the algebraic expression is: \[ 7x - 21y + 14z \]
Simplifying such expressions often involves factoring, which brings out a clearer structure.

This simplification process involves:
  • Identifying the greatest common factor
  • Factoring out the GCF
Doing so makes more complex expressions easier to manipulate and understand.

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