Select the correct answer.
Which of the following represents a factor from the expression given?
[tex]$5\left(3 x^2+9 x\right)-14$[/tex]
A. [tex]$15 x^2$[/tex]
B. [tex]$45 x$[/tex]
C. 5
D. 70
Answer (2)
None of the provided options are factors of the given expression. If forced to choose, 5 could be considered a factor of part of the original expression, but not the entire expression. There is likely an error in the question or the options provided.
Explanation
Understanding the Problem We are given the expression 5 ( 3 x 2 + 9 x ) − 14 and asked to identify which of the options 15 x 2 , 45 x , 5, and 70 is a factor of the expression.
Expanding the Expression First, let's expand the given expression: 5 ( 3 x 2 + 9 x ) − 14 = 15 x 2 + 45 x − 14
Checking Each Option Now, we will check each option to see if it's a factor of the expanded expression 15 x 2 + 45 x − 14 .
15 x 2 : For 15 x 2 to be a factor, we would need to write 15 x 2 + 45 x − 14 as 15 x 2 ( x ) for some function f ( x ) . This is not possible because the expression has terms 45 x and − 14 that cannot be obtained by multiplying 15 x 2 by a polynomial.
45 x : Similarly, for 45 x to be a factor, we would need to write 15 x 2 + 45 x − 14 as 45 x ( x ) for some function f ( x ) . This is not possible because the expression has terms 15 x 2 and − 14 that cannot be obtained by multiplying 45 x by a polynomial.
5 : For 5 to be a factor, we would need to be able to write 15 x 2 + 45 x − 14 as 5 ( x ) for some function h ( x ) . This is not possible since -14 is not divisible by 5.
70 : For 70 to be a factor, we would need to be able to write 15 x 2 + 45 x − 14 as 70 ( x ) for some function k ( x ) . This is not possible since the coefficients are not divisible by 70.
Re-examining the Problem Therefore, none of the given options are factors of the expression 15 x 2 + 45 x − 14 . However, since we are forced to select an answer, there must be an error in the question or the provided options. Let's re-examine the expression and options.
It seems there might be a misunderstanding of the term 'factor'. A factor usually refers to a term that divides the entire expression evenly. However, in this context, it seems the question is asking if any of the given terms appear in the expanded expression. In that case, 15 x 2 and 45 x are terms in the expanded expression 15 x 2 + 45 x − 14 . However, 5 and 70 are not.
Since the question asks for a 'factor', and none of the options are actual factors of the entire expression, the question is flawed. However, if we interpret the question as asking which of the given terms appear in the expanded expression, then 15 x 2 and 45 x would be valid answers. Since we must choose one, and 5 ( 3 x 2 + 9 x ) is part of the original expression, let's consider if 5 could be considered a factor of part of the expression. 5 ( 3 x 2 + 9 x ) does contain 5 as a factor. However, the -14 prevents 5 from being a factor of the entire expression.
Final Answer Given the constraints and the nature of the question, it seems there is no correct answer among the options provided. However, if we were forced to choose, the 'best' answer would be 5, as it is a factor of a part of the original expression before expansion. However, it is not a factor of the entire expression.
Conclusion Since none of the options are a factor of the entire expression, there is no correct answer. However, if we must choose one, we can say that 5 is a factor of part of the original expression.
Examples
Understanding factors is crucial in many real-world scenarios. For example, when designing a rectangular garden with an area represented by a quadratic expression, knowing the factors helps determine possible dimensions of the garden. Similarly, in physics, when analyzing the motion of an object, factors of the equation of motion can reveal key properties like initial velocity or acceleration. Factoring also plays a vital role in cryptography, where large numbers are factored to break encryption codes, highlighting its importance in data security.
The expression 5 ( 3 x 2 + 9 x ) − 14 expands to 15 x 2 + 45 x − 14 . Among the options given, none are factors of the entire expression, but option C, which is 5, can be considered a factor of part of the original expression before expansion. Therefore, if a selection must be made, the answer is C: 5 .
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