Write in factored form by factoring out the greatest common factor.
$24 s^3-8 s$
Answer (2)
Identify the greatest common factor (GCF) of the coefficients: The GCF of 24 and 8 is 8.
Identify the greatest common factor (GCF) of the variable terms: The GCF of s 3 and s is s .
Combine the GCFs: The overall GCF is 8 s .
Factor out the GCF from the expression: 24 s 3 − 8 s = 8 s ( 3 s 2 − 1 ) .
The factored form is 8 s ( 3 s 2 − 1 ) .
Explanation
Understanding the Problem We are asked to factor the expression 24 s 3 − 8 s by factoring out the greatest common factor (GCF). This means we need to identify the largest factor that divides both terms of the expression.
Finding the GCF of Coefficients First, let's find the GCF of the coefficients, which are 24 and 8. The greatest common factor of 24 and 8 is 8, since 8 divides both 24 and 8 without leaving a remainder.
Finding the GCF of Variables Next, let's find the GCF of the variable terms, which are s 3 and s . The greatest common factor of s 3 and s is s , since s is the highest power of s that divides both s 3 and s .
Determining the Overall GCF Now, we combine the GCF of the coefficients and the GCF of the variables to find the overall GCF of the expression. The GCF is 8 s .
Factoring out the GCF We factor out the GCF, 8 s , from each term in the expression: 24 s 3 − 8 s = 8 s ( 3 s 2 − 1 ) We divide each term in the original expression by 8 s to determine what remains inside the parentheses: 8 s 24 s 3 = 3 s 2 8 s − 8 s = − 1 So, the factored expression is 8 s ( 3 s 2 − 1 ) .
Final Answer Therefore, the factored form of the expression 24 s 3 − 8 s by factoring out the greatest common factor is 8 s ( 3 s 2 − 1 ) .
Examples
Factoring out the greatest common factor is a fundamental skill in algebra and is used in many real-world applications. For example, suppose you are designing a rectangular garden where the area is given by the expression 24 s 3 − 8 s , where s is a variable related to the dimensions of the garden. By factoring this expression as 8 s ( 3 s 2 − 1 ) , you can better understand the possible dimensions of the garden in terms of the common factor 8 s . This can help you optimize the layout and design of the garden, making it easier to manage and more aesthetically pleasing. Factoring simplifies complex expressions, making them easier to analyze and apply in practical situations.
The expression 24 s 3 − 8 s factors to 8 s ( 3 s 2 − 1 ) by finding the greatest common factor, which is 8 s . This GCF is derived from the coefficients 24 and 8, as well as the variables s 3 and s . The final factored form is useful for simplifying algebraic expressions.
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