Answer :
Let's tackle the long division of polynomials step by step: we need to divide [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] by [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex].
### Step-by-step Process:
1. Setup: Write down the dividend and the divisor.
- Dividend: [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex]
- Divisor: [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex]
2. Divide the Leading Terms:
- Take the leading term of the dividend [tex]\(-3x^5\)[/tex] and divide it 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] and subtract the result from the dividend:
[tex]\[
(-3x^2) \times (x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
- Subtract this from the dividend:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
[tex]\[
= 0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
4. Repeat the Process:
- Divide the new leading term [tex]\(-4x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
- Multiply and subtract similarly:
[tex]\[
(-4x) \times (x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
- Subtract:
[tex]\[
(0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
[tex]\[
= 0x^5 + 0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
5. Continue the Process:
- Divide the new leading term [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
- Multiply and subtract:
[tex]\[
(2) \times (x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
- Subtract:
[tex]\[
(0x^5 + 0x^4 + 2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
[tex]\[
= 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 4
\][/tex]
The result of the division, thus, is the quotient plus the remainder, which is:
[tex]\[
-3x^2 - 4x + 2 \quad \text{with a remainder of} \quad 4
\][/tex]
So, the final answer is:
[tex]\[
-3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
### Step-by-step Process:
1. Setup: Write down the dividend and the divisor.
- Dividend: [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex]
- Divisor: [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex]
2. Divide the Leading Terms:
- Take the leading term of the dividend [tex]\(-3x^5\)[/tex] and divide it 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] and subtract the result from the dividend:
[tex]\[
(-3x^2) \times (x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
- Subtract this from the dividend:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
[tex]\[
= 0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
4. Repeat the Process:
- Divide the new leading term [tex]\(-4x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
- Multiply and subtract similarly:
[tex]\[
(-4x) \times (x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
- Subtract:
[tex]\[
(0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
[tex]\[
= 0x^5 + 0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
5. Continue the Process:
- Divide the new leading term [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
- Multiply and subtract:
[tex]\[
(2) \times (x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
- Subtract:
[tex]\[
(0x^5 + 0x^4 + 2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
[tex]\[
= 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 4
\][/tex]
The result of the division, thus, is the quotient plus the remainder, which is:
[tex]\[
-3x^2 - 4x + 2 \quad \text{with a remainder of} \quad 4
\][/tex]
So, the final answer is:
[tex]\[
-3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
