Answer :
Sure, let's work through the polynomial division step-by-step using long division. We are dividing:
[tex]\[
-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6 \quad \text{by} \quad x^3 + 6x^2 - 3x - 5
\][/tex]
Step 1: Divide the leading terms
First, 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]
Step 2: Multiply and subtract
Multiply the entire divisor by [tex]\(-3x^2\)[/tex] and subtract this product from the original polynomial:
[tex]\[
(-3x^2) \cdot (x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
Subtract:
[tex]\[
\big(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\big) - \big(-3x^5 - 18x^4 + 9x^3 + 15x^2\big)
\][/tex]
This results in:
[tex]\[
0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
Step 3: Repeat the process
Divide the new leading term [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
Multiply the entire divisor by [tex]\(-4x\)[/tex] and subtract this product from the current polynomial:
[tex]\[
(-4x) \cdot (x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
Subtract:
[tex]\[
\big(-4x^4 - 22x^3 + 24x^2 + 14x - 6\big) - \big(-4x^4 - 24x^3 + 12x^2 + 20x\big)
\][/tex]
This results in:
[tex]\[
0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
Step 4: Repeat the process
Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
Multiply the entire divisor by [tex]\(2\)[/tex] and subtract this product from the current polynomial:
[tex]\[
2 \cdot (x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
Subtract:
[tex]\[
\big(2x^3 + 12x^2 - 6x - 6\big) - \big(2x^3 + 12x^2 - 6x - 10\big)
\][/tex]
This results in:
[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]. Thus, the division 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]
[tex]\[
-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6 \quad \text{by} \quad x^3 + 6x^2 - 3x - 5
\][/tex]
Step 1: Divide the leading terms
First, 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]
Step 2: Multiply and subtract
Multiply the entire divisor by [tex]\(-3x^2\)[/tex] and subtract this product from the original polynomial:
[tex]\[
(-3x^2) \cdot (x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]
Subtract:
[tex]\[
\big(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\big) - \big(-3x^5 - 18x^4 + 9x^3 + 15x^2\big)
\][/tex]
This results in:
[tex]\[
0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]
Step 3: Repeat the process
Divide the new leading term [tex]\(-4x^4\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]
Multiply the entire divisor by [tex]\(-4x\)[/tex] and subtract this product from the current polynomial:
[tex]\[
(-4x) \cdot (x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]
Subtract:
[tex]\[
\big(-4x^4 - 22x^3 + 24x^2 + 14x - 6\big) - \big(-4x^4 - 24x^3 + 12x^2 + 20x\big)
\][/tex]
This results in:
[tex]\[
0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]
Step 4: Repeat the process
Divide [tex]\(2x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]
Multiply the entire divisor by [tex]\(2\)[/tex] and subtract this product from the current polynomial:
[tex]\[
2 \cdot (x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]
Subtract:
[tex]\[
\big(2x^3 + 12x^2 - 6x - 6\big) - \big(2x^3 + 12x^2 - 6x - 10\big)
\][/tex]
This results in:
[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]. Thus, the division 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]
