Question
what is a common factor for the expression 24x^2 + 16x + 144
A. 16
B. 8x
C. 3x^2 +2x +18
D. 8(x-2)(3x^2 +9)
A. 16
B. 8x
C. 3x^2 +2x +18
D. 8(x-2)(3x^2 +9)
Answers
GPT-4o mini
To find a common factor for the expression \(24x^2 + 16x + 144\), we can start by factoring out the greatest common factor (GCF) from all the terms.
1. Identify the GCF of the coefficients 24, 16, and 144:
- The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
- The factors of 16 are 1, 2, 4, 8, 16.
- The factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144.
- The common factors are 1, 2, 4, 8.
- The GCF is 8.
2. Now we can factor the expression by dividing each term by 8:
\[
24x^2 + 16x + 144 = 8(3x^2 + 2x + 18).
\]
Now we can evaluate the answer choices you provided:
A. 16 - This is **not** a common factor, \(8\) is greater.
B. 8x - This cannot be a common factor since \(8x\) does not divide all terms.
C. \(3x^2 + 2x + 18\) - This is the expression factored out from \(8\), but it's not a common factor.
D. \(8(x - 2)(3x^2 + 9)\) - This isn't equivalent to the original expression when expanded.
Thus, the strongest common factor from the options given is **8**. Since it's not listed, the best option based off my findings is \(C\) but keep in mind that \(8\) is the actual common factor.
So the correct response based on your list would be dependent on the context of the question. If you require the direct common factor, it would be \(8\). If you are strictly choosing from the options provided with the functioning context, C or D might lean in close idealism but aren't precisely concise on their own.
1. Identify the GCF of the coefficients 24, 16, and 144:
- The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
- The factors of 16 are 1, 2, 4, 8, 16.
- The factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144.
- The common factors are 1, 2, 4, 8.
- The GCF is 8.
2. Now we can factor the expression by dividing each term by 8:
\[
24x^2 + 16x + 144 = 8(3x^2 + 2x + 18).
\]
Now we can evaluate the answer choices you provided:
A. 16 - This is **not** a common factor, \(8\) is greater.
B. 8x - This cannot be a common factor since \(8x\) does not divide all terms.
C. \(3x^2 + 2x + 18\) - This is the expression factored out from \(8\), but it's not a common factor.
D. \(8(x - 2)(3x^2 + 9)\) - This isn't equivalent to the original expression when expanded.
Thus, the strongest common factor from the options given is **8**. Since it's not listed, the best option based off my findings is \(C\) but keep in mind that \(8\) is the actual common factor.
So the correct response based on your list would be dependent on the context of the question. If you require the direct common factor, it would be \(8\). If you are strictly choosing from the options provided with the functioning context, C or D might lean in close idealism but aren't precisely concise on their own.