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x^3 - 2x - 12x^3 - 48 + 96x + 384 divided by (x - 9)

Answer :

To divide the polynomial [tex]x^3 - 2x - 12x^3 - 48 + 96x + 384[/tex] by [tex]x - 9[/tex], we should first simplify and rearrange the terms properly.

Step 1: Combine like terms.

The original expression [tex]x^3 - 2x - 12x^3 - 48 + 96x + 384[/tex] can be simplified by combining like terms:

[tex](x^3 - 12x^3) + (-2x + 96x) - 48 + 384[/tex]

This simplifies to:

[tex]-11x^3 + 94x + 336[/tex]

Step 2: Perform polynomial long division.

Now, divide [tex]-11x^3 + 94x + 336[/tex] by [tex]x - 9[/tex] using polynomial long division.

  1. Divide the leading term [tex]-11x^3[/tex] by the leading term of the divisor [tex]x[/tex]:
    [tex]-11x^2[/tex].

  2. Multiply [tex]-11x^2[/tex] by [tex]x - 9[/tex]:
    [tex]-11x^3 + 99x^2[/tex].

  3. Subtract [tex]-11x^3 + 99x^2[/tex] from [tex]-11x^3 + 94x + 336[/tex]:

    [tex]-11x^3 + 94x + 336[/tex]

    • [tex]-11x^3 + 99x^2[/tex]

    [tex]- 99x^2 + 94x + 336[/tex].

    Note in this step, an error in placing of terms will help check if my subtraction resulted in the proper terms lined up properly.

  4. Bring down the next term, in this case, we already have all terms accounted for.

  5. Divide [tex]-99x^2[/tex] by [tex]x[/tex]:
    [tex]-99x[/tex].

  6. Multiply [tex]-99x[/tex] by [tex]x - 9[/tex]:
    [tex]-99x^2 + 891x[/tex].

  7. Subtract [tex]-99x^2 + 891x[/tex] from [tex]-99x^2 + 94x + 336[/tex]:

    [tex]-99x^2 + 94x + 336[/tex]
    -[tex]-99x^2 + 891x[/tex]

    [tex]-797x + 336[/tex].

  8. Repeat for the last time: divide [tex]-797x[/tex] by [tex]x[/tex]:
    [tex]-797[/tex].

  9. Multiply [tex]-797[/tex] by [tex]x - 9[/tex]:
    [tex]-797x + 7173[/tex].

  10. Subtract [tex]-797x + 7173[/tex] from [tex]-797x + 336[/tex]:

    [tex]-797x + 336[/tex]
    -[tex]-797x + 7173[/tex]

    [tex]-6837[/tex].

Since [tex]-6837[/tex] is the remainder, the final result can be expressed as:

[tex]-11x^2 - 99x - 797 + \frac{-6837}{x - 9}[/tex]

Polynomial division involves these types of calculations that closely models the long division you learn with arithmetic, but applies it to these algebraic expressions. The reminder [tex]-6837/(x-9)[/tex] indicates that if you were looking for complete clean division, we don't quite have it, but this format is a meaningful way to express what remains after the main terms have been divided.

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Rewritten by : Barada