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Chapter 1: Problem 156

Find an equivalent expression by factoring. $$ 6 y-9 x-3 w $$

Short Answer

Expert verified
3(2y - 3x - w)

Step by step solution

01

- Identify Common Factors

The first step in factoring an expression is to identify the common factor in each term. The terms in the given expression are 6y, -9x, and -3w. All these terms have a common factor of 3.
02

- Factor Out the Common Factor

Next, factor out the common factor from each term. The expression can be rewritten by factoring out 3: \[ 6y - 9x - 3w = 3(2y - 3x - w) \]
03

- Verify the Factored Expression

Lastly, verify if the expression inside the parenthesis is accurate by distributing the 3 back to each term: \[ 3(2y - 3x - w) = 3 \times 2y + 3 \times (-3x) + 3 \times (-w) = 6y - 9x - 3w \]. The factored form is correct.

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Key Concepts

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

common factors
Understanding common factors is the first step in the factoring process. A common factor is a number or variable that is a factor of each term in an expression. Identifying common factors makes it easier to simplify expressions and solve equations. For the given problem, which is factoring the expression \(6y - 9x - 3w\), it鈥檚 essential to look for a number that can divide 6, -9, and -3 completely. In this case, all these numbers share a common factor of 3. This means each term in the expression can be divided by 3 without leaving a remainder. Identifying such commonalities simplifies further operations and is crucial for successful factoring.
factoring process
The factoring process involves rewriting an expression as a product of its factors. After identifying the common factor, the next step is to factor it out from each term in the expression. In the example, the common factor is 3. To factor out 3, you divide each term by 3 and place the resulting expression inside parentheses: \[ 6y - 9x - 3w = 3(2y - 3x - w) \]. This process simplifies the given expression by reducing it to a product of the common factor and another simpler expression, making it easier to work with in subsequent steps.
verifying factored expressions
Verification is a crucial step in the factoring process to ensure the factored form is accurate. You can verify if the factored expression matches the original by distributing the factored term back through the parentheses. In the problem, the factored expression is \(3(2y - 3x - w)\). To verify, distribute 3: \[ 3(2y - 3x - w) = 3 \times 2y + 3 \times (-3x) + 3 \times (-w) = 6y - 9x - 3w \]. This matches the original expression, confirming that the factoring was done correctly. Verification helps ensure no mistakes were made during the factoring process and instills confidence in the solution.

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