Answer :
To find which term can be used in the blank of [tex]\(36x^3 - 22x^2 - \underline{\ \ \ }\)[/tex] so that the greatest common factor (GCF) of the resulting polynomial is [tex]\(2x\)[/tex], we need to analyze each term option and determine their impact on the GCF of the polynomial.
We'll consider the polynomial [tex]\(36x^3 - 22x^2 - ax\)[/tex] or [tex]\(36x^3 - 22x^2 - a\)[/tex], depending on the form of the term. We aim to ensure that the GCF of the polynomial includes [tex]\(2x\)[/tex].
### Analyzing Each Term
1. Term: [tex]\(4xy\)[/tex]
[tex]\[ Poly = 36x^3 - 22x^2 - 4xy \][/tex]
- GCF of [tex]\(36x^3\)[/tex], [tex]\(22x^2\)[/tex], and [tex]\(4xy\)[/tex]:
- The coefficients [tex]\(36, 22,\)[/tex] and [tex]\(4\)[/tex] have a common factor of [tex]\(2\)[/tex].
- The variables have a common factor of [tex]\(x\)[/tex], as the polynomial terms are [tex]\(x^3, x^2,\)[/tex] and [tex]\(xy\)[/tex].
Thus, the GCF is [tex]\(2x\)[/tex].
2. Term: [tex]\(12x\)[/tex]
[tex]\[ Poly = 36x^3 - 22x^2 - 12x \][/tex]
- GCF of [tex]\(36x^3\)[/tex], [tex]\(22x^2\)[/tex], and [tex]\(12x\)[/tex]:
- The coefficients [tex]\(36, 22,\)[/tex] and [tex]\(12\)[/tex] have a common factor of [tex]\(2\)[/tex].
- The variables all have at least an [tex]\(x\)[/tex] term.
Thus, the GCF is [tex]\(2x\)[/tex].
3. Term: [tex]\(24\)[/tex]
[tex]\[ Poly = 36x^3 - 22x^2 - 24 \][/tex]
- GCF of [tex]\(36x^3\)[/tex], [tex]\(22x^2\)[/tex], and [tex]\(24\)[/tex]:
- The coefficients [tex]\(36, 22,\)[/tex] and [tex]\(24\)[/tex] have a common factor of [tex]\(2\)[/tex].
- The third term, [tex]\(24\)[/tex], does not include an [tex]\(x\)[/tex] factor.
Thus, there is no common [tex]\(x\)[/tex] term in all polynomial terms, so the GCF would be only [tex]\(2\)[/tex].
4. Term: [tex]\(44y\)[/tex]
[tex]\[ Poly = 36x^3 - 22x^2 - 44y \][/tex]
- GCF of [tex]\(36x^3\)[/tex], [tex]\(22x^2\)[/tex], and [tex]\(44y\)[/tex]:
- The coefficients [tex]\(36, 22,\)[/tex] and [tex]\(44\)[/tex] have a common factor of [tex]\(2\)[/tex].
- The variables have no common factor; the third term has [tex]\(y\)[/tex] and not [tex]\(x\)[/tex].
Thus, there is no common [tex]\(x\)[/tex] term in all polynomial terms, so the GCF would be only [tex]\(2\)[/tex].
### Conclusion:
Evaluating the terms, we find that the GCF of the polynomial [tex]\(36x^3 - 22x^2 - \underline{\ \ \ }\)[/tex] includes [tex]\(2x\)[/tex] when the terms [tex]\(4xy\)[/tex] and [tex]\(12x\)[/tex] are used.
#### Therefore, the correct options are:
- [tex]\(4xy\)[/tex]
- [tex]\(12x\)[/tex]
We'll consider the polynomial [tex]\(36x^3 - 22x^2 - ax\)[/tex] or [tex]\(36x^3 - 22x^2 - a\)[/tex], depending on the form of the term. We aim to ensure that the GCF of the polynomial includes [tex]\(2x\)[/tex].
### Analyzing Each Term
1. Term: [tex]\(4xy\)[/tex]
[tex]\[ Poly = 36x^3 - 22x^2 - 4xy \][/tex]
- GCF of [tex]\(36x^3\)[/tex], [tex]\(22x^2\)[/tex], and [tex]\(4xy\)[/tex]:
- The coefficients [tex]\(36, 22,\)[/tex] and [tex]\(4\)[/tex] have a common factor of [tex]\(2\)[/tex].
- The variables have a common factor of [tex]\(x\)[/tex], as the polynomial terms are [tex]\(x^3, x^2,\)[/tex] and [tex]\(xy\)[/tex].
Thus, the GCF is [tex]\(2x\)[/tex].
2. Term: [tex]\(12x\)[/tex]
[tex]\[ Poly = 36x^3 - 22x^2 - 12x \][/tex]
- GCF of [tex]\(36x^3\)[/tex], [tex]\(22x^2\)[/tex], and [tex]\(12x\)[/tex]:
- The coefficients [tex]\(36, 22,\)[/tex] and [tex]\(12\)[/tex] have a common factor of [tex]\(2\)[/tex].
- The variables all have at least an [tex]\(x\)[/tex] term.
Thus, the GCF is [tex]\(2x\)[/tex].
3. Term: [tex]\(24\)[/tex]
[tex]\[ Poly = 36x^3 - 22x^2 - 24 \][/tex]
- GCF of [tex]\(36x^3\)[/tex], [tex]\(22x^2\)[/tex], and [tex]\(24\)[/tex]:
- The coefficients [tex]\(36, 22,\)[/tex] and [tex]\(24\)[/tex] have a common factor of [tex]\(2\)[/tex].
- The third term, [tex]\(24\)[/tex], does not include an [tex]\(x\)[/tex] factor.
Thus, there is no common [tex]\(x\)[/tex] term in all polynomial terms, so the GCF would be only [tex]\(2\)[/tex].
4. Term: [tex]\(44y\)[/tex]
[tex]\[ Poly = 36x^3 - 22x^2 - 44y \][/tex]
- GCF of [tex]\(36x^3\)[/tex], [tex]\(22x^2\)[/tex], and [tex]\(44y\)[/tex]:
- The coefficients [tex]\(36, 22,\)[/tex] and [tex]\(44\)[/tex] have a common factor of [tex]\(2\)[/tex].
- The variables have no common factor; the third term has [tex]\(y\)[/tex] and not [tex]\(x\)[/tex].
Thus, there is no common [tex]\(x\)[/tex] term in all polynomial terms, so the GCF would be only [tex]\(2\)[/tex].
### Conclusion:
Evaluating the terms, we find that the GCF of the polynomial [tex]\(36x^3 - 22x^2 - \underline{\ \ \ }\)[/tex] includes [tex]\(2x\)[/tex] when the terms [tex]\(4xy\)[/tex] and [tex]\(12x\)[/tex] are used.
#### Therefore, the correct options are:
- [tex]\(4xy\)[/tex]
- [tex]\(12x\)[/tex]
