Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.

If the equation [tex]$2x^3 - 9x^2 - 6 = 0$[/tex] is transformed into an equation in which the second term is missing, the roots are to be diminished by

A. [tex]$2/3$[/tex]

B. [tex][tex]$-2/3$[/tex][/tex]

C. [tex]$3/2$[/tex]

D. [tex]$-3/2$[/tex]


Sagot :

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]