Answer :
Sure! Let's solve this polynomial division step-by-step using long division.
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 1: Set up the division
Write the division as a long division problem, with 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.
### Step 2: Divide the leading terms
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.
### Step 3: Multiply and subtract
Multiply the entire divisor [tex]\((x^3 + 6x^2 - 3x - 5)\)[/tex] by [tex]\(-3x^2\)[/tex], and subtract this from the current dividend:
1. [tex]\(-3x^2 \times x^3 = -3x^5\)[/tex]
2. [tex]\(-3x^2 \times 6x^2 = -18x^4\)[/tex]
3. [tex]\(-3x^2 \times (-3x) = 9x^3\)[/tex]
4. [tex]\(-3x^2 \times (-5) = 15x^2\)[/tex]
Now perform the subtraction:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
This simplifies to:
[tex]\[
0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
### Step 4: Repeat the process
Now, divide the new leading term [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 divisor by [tex]\(-4x\)[/tex], and subtract:
1. [tex]\(-4x \times x^3 = -4x^4\)[/tex]
2. [tex]\(-4x \times 6x^2 = -24x^3\)[/tex]
3. [tex]\(-4x \times (-3x) = 12x^2\)[/tex]
4. [tex]\(-4x \times (-5) = 20x\)[/tex]
Subtract:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
Simplifies to:
[tex]\[
0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
### Step 5: Divide again
Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
This is the next term of the quotient.
Multiply the divisor by 2, and subtract:
1. [tex]\(2 \times x^3 = 2x^3\)[/tex]
2. [tex]\(2 \times 6x^2 = 12x^2\)[/tex]
3. [tex]\(2 \times (-3x) = -6x\)[/tex]
4. [tex]\(2 \times (-5) = -10\)[/tex]
Subtract:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
Simplifies to:
[tex]\[
0x^3 + 0x^2 + 0x + 4
\][/tex]
### Conclusion
The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(4\)[/tex].
So, the result of the division is:
[tex]\[
-3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
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 1: Set up the division
Write the division as a long division problem, with 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.
### Step 2: Divide the leading terms
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.
### Step 3: Multiply and subtract
Multiply the entire divisor [tex]\((x^3 + 6x^2 - 3x - 5)\)[/tex] by [tex]\(-3x^2\)[/tex], and subtract this from the current dividend:
1. [tex]\(-3x^2 \times x^3 = -3x^5\)[/tex]
2. [tex]\(-3x^2 \times 6x^2 = -18x^4\)[/tex]
3. [tex]\(-3x^2 \times (-3x) = 9x^3\)[/tex]
4. [tex]\(-3x^2 \times (-5) = 15x^2\)[/tex]
Now perform the subtraction:
[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]
This simplifies to:
[tex]\[
0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
### Step 4: Repeat the process
Now, divide the new leading term [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 divisor by [tex]\(-4x\)[/tex], and subtract:
1. [tex]\(-4x \times x^3 = -4x^4\)[/tex]
2. [tex]\(-4x \times 6x^2 = -24x^3\)[/tex]
3. [tex]\(-4x \times (-3x) = 12x^2\)[/tex]
4. [tex]\(-4x \times (-5) = 20x\)[/tex]
Subtract:
[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]
Simplifies to:
[tex]\[
0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
### Step 5: Divide again
Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
This is the next term of the quotient.
Multiply the divisor by 2, and subtract:
1. [tex]\(2 \times x^3 = 2x^3\)[/tex]
2. [tex]\(2 \times 6x^2 = 12x^2\)[/tex]
3. [tex]\(2 \times (-3x) = -6x\)[/tex]
4. [tex]\(2 \times (-5) = -10\)[/tex]
Subtract:
[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]
Simplifies to:
[tex]\[
0x^3 + 0x^2 + 0x + 4
\][/tex]
### Conclusion
The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(4\)[/tex].
So, the result of the division is:
[tex]\[
-3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]
