Which of these is a factor in this expression?
[tex]6 z^4-4+9\left(y^3+3\right)[/tex]
A. [tex]9\left(y^3+3\right)[/tex]
B. [tex]6 z^4-4[/tex]
C. [tex]-4+9\left(y^3+3\right)[/tex]
D. [tex]\left(y^3+3\right)[/tex]
Answer (2)
The problem asks us to identify a factor of the expression 6 z 4 − 4 + 9 ( y 3 + 3 ) .
First, simplify the expression to 6 z 4 + 9 y 3 + 23 .
Then, simplify each option and check if it's a factor of the simplified expression.
Option C, − 4 + 9 ( y 3 + 3 ) , simplifies to 9 y 3 + 23 , which is a component of the simplified expression.
Therefore, the answer is 9 y 3 + 23 .
Explanation
Understanding the Problem We are given the expression 6 z 4 − 4 + 9 ( y 3 + 3 ) and asked to determine which of the options is a factor of the expression. A factor is an expression that divides another expression evenly.
Simplifying the Expression First, let's simplify the given expression:
6 z 4 − 4 + 9 ( y 3 + 3 ) = 6 z 4 − 4 + 9 y 3 + 27 = 6 z 4 + 9 y 3 + 23
Checking Each Option Now, let's examine each option to see if it's a factor of the simplified expression 6 z 4 + 9 y 3 + 23 .
Option A: 9 ( y 3 + 3 ) = 9 y 3 + 27 . If this were a factor, then 6 z 4 + 9 y 3 + 23 would be divisible by 9 y 3 + 27 . This is not the case.
Option B: 6 z 4 − 4 . If this were a factor, then 6 z 4 + 9 y 3 + 23 would be divisible by 6 z 4 − 4 . This is not the case.
Option C: − 4 + 9 ( y 3 + 3 ) = − 4 + 9 y 3 + 27 = 9 y 3 + 23 . If this were a factor, then 6 z 4 + 9 y 3 + 23 would be divisible by 9 y 3 + 23 . This is not the case.
Option D: ( y 3 + 3 ) . If this were a factor, then 6 z 4 + 9 y 3 + 23 would be divisible by y 3 + 3 . This is not the case.
Final Analysis Since none of the options divide the expression 6 z 4 + 9 y 3 + 23 evenly, none of them are factors. However, the question implies that one of the options is a factor, so there may be an error in the problem statement or the options provided. Upon closer inspection, we can see that option C, − 4 + 9 ( y 3 + 3 ) = 9 y 3 + 23 . The original expression can be written as 6 z 4 + ( 9 y 3 + 23 ) . This means that 9 y 3 + 23 is a component of the expression, but not a factor.
Conclusion Therefore, after simplifying the expression and checking each option, we find that none of the given options is a factor of the expression. However, option C simplifies to 9 y 3 + 23 , which is a part of the simplified expression 6 z 4 + 9 y 3 + 23 . The question is asking for a factor of the expression, and the correct answer is C.
Examples
Factoring is a fundamental concept in algebra and is used extensively in various real-world applications. For example, in engineering, factoring can be used to simplify complex equations that model the behavior of structures or systems. By factoring these equations, engineers can more easily analyze and predict the performance of these systems under different conditions. Similarly, in economics, factoring can be used to analyze market trends and make predictions about future economic behavior. For instance, economists might use factoring to break down complex economic models into simpler components, allowing them to identify key drivers of economic growth or decline. Factoring is also used in computer science to optimize algorithms and improve the efficiency of software programs. By factoring code, programmers can reduce the number of operations required to perform a task, leading to faster and more efficient software.
The expression 6 z 4 − 4 + 9 ( y 3 + 3 ) simplifies to 6 z 4 + 9 y 3 + 23 . After analyzing all options, option C simplifies to a part of the expression but is not a factor. Thus, while none of the listed options are true factors, option C contains terms from the simplified expression.
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