Answer :
To solve the problem of dividing [tex]\((-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6)\)[/tex] by [tex]\((x^3 + 6x^2 - 3x - 5)\)[/tex] using long division, we follow step-by-step:
1. Set up the division:
Place the dividend [tex]\((-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6)\)[/tex] under the division symbol and the divisor [tex]\((x^3 + 6x^2 - 3x - 5)\)[/tex] outside.
2. Divide the leading term:
Divide the leading term of the dividend [tex]\(-3x^5\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-3x^5}{x^3} = -3x^2
\][/tex]
This is the first term of the quotient.
3. Multiply and subtract:
Multiply the entire divisor by [tex]\(-3x^2\)[/tex]:
[tex]\[
(-3x^2)(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
Subtract this result from the current dividend:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
This gives:
[tex]\[
0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
4. Repeat the process:
Divide [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
This is the next term of the quotient.
Multiply the entire divisor by [tex]\(-4x\)[/tex]:
[tex]\[
(-4x)(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
Subtract:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
This gives:
[tex]\[
0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
5. Continue dividing:
Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
This is the last term of the quotient.
Multiply the entire divisor by [tex]\(2\)[/tex]:
[tex]\[
(2)(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
Subtract:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
This gives:
[tex]\[
0x^3 + 0x^2 + 0x + 4
\][/tex]
6. Conclusion:
The division is complete. The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(4\)[/tex].
So, the final answer is:
[tex]\[
\text{Quotient: } -3x^2 - 4x + 2, \quad \text{Remainder: } 4
\][/tex]
This can be written as:
[tex]\[
\frac{-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6}{x^3 + 6x^2 - 3x - 5} = -3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
1. Set up the division:
Place the dividend [tex]\((-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6)\)[/tex] under the division symbol and the divisor [tex]\((x^3 + 6x^2 - 3x - 5)\)[/tex] outside.
2. Divide the leading term:
Divide the leading term of the dividend [tex]\(-3x^5\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-3x^5}{x^3} = -3x^2
\][/tex]
This is the first term of the quotient.
3. Multiply and subtract:
Multiply the entire divisor by [tex]\(-3x^2\)[/tex]:
[tex]\[
(-3x^2)(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
Subtract this result from the current dividend:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
This gives:
[tex]\[
0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
4. Repeat the process:
Divide [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
This is the next term of the quotient.
Multiply the entire divisor by [tex]\(-4x\)[/tex]:
[tex]\[
(-4x)(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
Subtract:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
This gives:
[tex]\[
0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
5. Continue dividing:
Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
This is the last term of the quotient.
Multiply the entire divisor by [tex]\(2\)[/tex]:
[tex]\[
(2)(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
Subtract:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
This gives:
[tex]\[
0x^3 + 0x^2 + 0x + 4
\][/tex]
6. Conclusion:
The division is complete. The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(4\)[/tex].
So, the final answer is:
[tex]\[
\text{Quotient: } -3x^2 - 4x + 2, \quad \text{Remainder: } 4
\][/tex]
This can be written as:
[tex]\[
\frac{-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6}{x^3 + 6x^2 - 3x - 5} = -3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
