Answer :
Of course! Let's divide the polynomial [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] by [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] using long division, step-by-step.
### Step 1: Set up the division
Write the division in long division format. The dividend is [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] and the divisor is [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex].
### Step 2: Divide the leading terms
Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{-3x^5}{x^3} = -3x^2
\][/tex]
This will be the first term of the quotient.
### Step 3: Multiply and subtract
Multiply the entire divisor by [tex]\(-3x^2\)[/tex] and subtract the result from the dividend:
[tex]\[
(-3x^2)(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
Subtract from the 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]
### Step 4: Repeat the process
Now take this result as the new dividend and repeat the process.
#### Divide the leading terms:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
Add [tex]\(-4x\)[/tex] to the quotient.
#### Multiply and subtract:
[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]
### Step 5: Divide again
#### Divide the leading terms:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
Add [tex]\(2\)[/tex] to the quotient.
#### Multiply and subtract:
[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 leaves:
[tex]\[
0x^3 + 0x^2 + 0x + 4 \quad \text{(remainder)}
\][/tex]
### Conclusion
The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(4\)[/tex]. Therefore, the division can be expressed 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]
### Step 1: Set up the division
Write the division in long division format. The dividend is [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] and the divisor is [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex].
### Step 2: Divide the leading terms
Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{-3x^5}{x^3} = -3x^2
\][/tex]
This will be the first term of the quotient.
### Step 3: Multiply and subtract
Multiply the entire divisor by [tex]\(-3x^2\)[/tex] and subtract the result from the dividend:
[tex]\[
(-3x^2)(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
Subtract from the 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]
### Step 4: Repeat the process
Now take this result as the new dividend and repeat the process.
#### Divide the leading terms:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
Add [tex]\(-4x\)[/tex] to the quotient.
#### Multiply and subtract:
[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]
### Step 5: Divide again
#### Divide the leading terms:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
Add [tex]\(2\)[/tex] to the quotient.
#### Multiply and subtract:
[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 leaves:
[tex]\[
0x^3 + 0x^2 + 0x + 4 \quad \text{(remainder)}
\][/tex]
### Conclusion
The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(4\)[/tex]. Therefore, the division can be expressed 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]
