Answer :
To transform the given polynomial [tex]\(2x^3 - 9x^2 - 6 = 0\)[/tex] into a form where the [tex]\(x^2\)[/tex] term is removed, we need to substitute [tex]\(x\)[/tex] with [tex]\(y + a\)[/tex], where [tex]\(a\)[/tex] is a specific value that will zero out the [tex]\(x^2\)[/tex] term.
### Steps
1. Substitute [tex]\(x\)[/tex] with [tex]\(y + a\)[/tex]:
[tex]\[ x = y + a \][/tex]
Substituting [tex]\(x = y + a\)[/tex] into the polynomial [tex]\(2x^3 - 9x^2 - 6\)[/tex]:
[tex]\[ 2(y + a)^3 - 9(y + a)^2 - 6 \][/tex]
2. Expand [tex]\(2(y + a)^3\)[/tex]:
[tex]\[ 2(y^3 + 3ay^2 + 3a^2y + a^3) = 2y^3 + 6ay^2 + 6a^2y + 2a^3 \][/tex]
3. Expand [tex]\(-9(y + a)^2\)[/tex]:
[tex]\[ -9(y^2 + 2ay + a^2) = -9y^2 - 18ay - 9a^2 \][/tex]
4. Combine all terms:
[tex]\[ 2y^3 + 6ay^2 + 6a^2y + 2a^3 - 9y^2 - 18ay - 9a^2 - 6 \][/tex]
5. Group terms with similar powers of [tex]\(y\)[/tex]:
[tex]\[ 2y^3 + (6a - 9)y^2 + (6a^2 - 18a)y + (2a^3 - 9a^2 - 6) \][/tex]
6. Set the coefficient of [tex]\(y^2\)[/tex] to zero:
[tex]\[ 6a - 9 = 0 \][/tex]
Solve for [tex]\(a\)[/tex]:
[tex]\[ 6a = 9 \][/tex]
[tex]\[ a = \frac{9}{6} \][/tex]
[tex]\[ a = \frac{3}{2} \][/tex]
Therefore, the roots are diminished by [tex]\(\frac{3}{2}\)[/tex].
### Conclusion
The correct option is:
C) [tex]\(\frac{3}{2}\)[/tex]
### Steps
1. Substitute [tex]\(x\)[/tex] with [tex]\(y + a\)[/tex]:
[tex]\[ x = y + a \][/tex]
Substituting [tex]\(x = y + a\)[/tex] into the polynomial [tex]\(2x^3 - 9x^2 - 6\)[/tex]:
[tex]\[ 2(y + a)^3 - 9(y + a)^2 - 6 \][/tex]
2. Expand [tex]\(2(y + a)^3\)[/tex]:
[tex]\[ 2(y^3 + 3ay^2 + 3a^2y + a^3) = 2y^3 + 6ay^2 + 6a^2y + 2a^3 \][/tex]
3. Expand [tex]\(-9(y + a)^2\)[/tex]:
[tex]\[ -9(y^2 + 2ay + a^2) = -9y^2 - 18ay - 9a^2 \][/tex]
4. Combine all terms:
[tex]\[ 2y^3 + 6ay^2 + 6a^2y + 2a^3 - 9y^2 - 18ay - 9a^2 - 6 \][/tex]
5. Group terms with similar powers of [tex]\(y\)[/tex]:
[tex]\[ 2y^3 + (6a - 9)y^2 + (6a^2 - 18a)y + (2a^3 - 9a^2 - 6) \][/tex]
6. Set the coefficient of [tex]\(y^2\)[/tex] to zero:
[tex]\[ 6a - 9 = 0 \][/tex]
Solve for [tex]\(a\)[/tex]:
[tex]\[ 6a = 9 \][/tex]
[tex]\[ a = \frac{9}{6} \][/tex]
[tex]\[ a = \frac{3}{2} \][/tex]
Therefore, the roots are diminished by [tex]\(\frac{3}{2}\)[/tex].
### Conclusion
The correct option is:
C) [tex]\(\frac{3}{2}\)[/tex]
