Answer :
- Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient: $-3x^5 / x^3 = -3x^2$.
- Multiply the divisor by the first term of the quotient and subtract the result from the dividend: $(-3x^5-22x^4-13x^3+39x^2+14x-6) - (-3x^2)(x^3+6x^2-3x-5) = -4x^4 - 22x^3 + 24x^2 + 14x - 6$.
- Repeat the process to find the remaining terms of the quotient and the remainder: $-4x^4 / x^3 = -4x$, $( -4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x)(x^3+6x^2-3x-5) = 2x^3 + 12x^2 - 6x - 6$, $2x^3 / x^3 = 2$, $(2x^3 + 12x^2 - 6x - 6) - (2)(x^3+6x^2-3x-5) = 4$.
- The quotient is $-3x^2 - 4x + 2$ and the remainder is $4$, so the final answer is $\boxed{-3x^2 - 4x + 2 + \frac{4}{x^3+6x^2-3x-5}}$.
### Explanation
1. Understanding the Problem
We are asked to perform polynomial long division to divide $-3 x^5-22 x^4-13 x^3+39 x^2+14 x-6$ by $x^3+6 x^2-3 x-5$. This means we want to find polynomials $q(x)$ (the quotient) and $r(x)$ (the remainder) such that $-3 x^5-22 x^4-13 x^3+39 x^2+14 x-6 = (x^3+6 x^2-3 x-5)q(x) + r(x)$, where the degree of $r(x)$ is less than the degree of $x^3+6 x^2-3 x-5$, which is 3.
2. Finding the First Term of the Quotient
We start by dividing the highest degree term of the dividend, $-3x^5$, by the highest degree term of the divisor, $x^3$. This gives us $-3x^2$, which is the first term of the quotient.
3. Subtracting and Bringing Down
Next, we multiply the divisor, $x^3+6 x^2-3 x-5$, by $-3x^2$ to get $-3x^5 - 18x^4 + 9x^3 + 15x^2$. We subtract this from the dividend: $(-3 x^5-22 x^4-13 x^3+39 x^2+14 x-6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2) = -4x^4 - 22x^3 + 24x^2 + 14x - 6$.
4. Finding the Second Term of the Quotient
Now we repeat the process with the new dividend, $-4x^4 - 22x^3 + 24x^2 + 14x - 6$. We divide the highest degree term, $-4x^4$, by the highest degree term of the divisor, $x^3$, which gives us $-4x$. This is the next term of the quotient.
5. Subtracting Again
We multiply the divisor, $x^3+6 x^2-3 x-5$, by $-4x$ to get $-4x^4 - 24x^3 + 12x^2 + 20x$. We subtract this from the new dividend: $(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x) = 2x^3 + 12x^2 - 6x - 6$.
6. Finding the Third Term of the Quotient
We repeat the process again with the new dividend, $2x^3 + 12x^2 - 6x - 6$. We divide the highest degree term, $2x^3$, by the highest degree term of the divisor, $x^3$, which gives us $2$. This is the next term of the quotient.
7. Final Subtraction
We multiply the divisor, $x^3+6 x^2-3 x-5$, by $2$ to get $2x^3 + 12x^2 - 6x - 10$. We subtract this from the new dividend: $(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10) = 4$.
8. Determining the Quotient and Remainder
The remainder is $4$, which has a degree of 0, less than the degree of the divisor, which is 3. Therefore, the quotient is $-3x^2 - 4x + 2$ and the remainder is $4$.
9. Final Answer
Thus, $\frac{-3 x^5-22 x^4-13 x^3+39 x^2+14 x-6}{x^3+6 x^2-3 x-5} = -3x^2 - 4x + 2 + \frac{4}{x^3+6 x^2-3 x-5}$. The quotient is $-3x^2 - 4x + 2$ and the remainder is $4$.
10. Conclusion
The result of the long division is a quotient of $-3x^2 - 4x + 2$ and a remainder of $4$. Therefore, the answer is $-3x^2 - 4x + 2 + \frac{4}{x^3+6x^2-3x-5}$.
### Examples
Polynomial long division is used in various engineering and scientific applications. For instance, in control systems, it helps simplify transfer functions to analyze system stability and response. In signal processing, it can decompose complex signals into simpler components for easier analysis and manipulation. Moreover, in computer graphics, polynomial division can be used to optimize rendering algorithms by simplifying complex polynomial expressions that define curves and surfaces, leading to faster and more efficient graphics processing.
- Multiply the divisor by the first term of the quotient and subtract the result from the dividend: $(-3x^5-22x^4-13x^3+39x^2+14x-6) - (-3x^2)(x^3+6x^2-3x-5) = -4x^4 - 22x^3 + 24x^2 + 14x - 6$.
- Repeat the process to find the remaining terms of the quotient and the remainder: $-4x^4 / x^3 = -4x$, $( -4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x)(x^3+6x^2-3x-5) = 2x^3 + 12x^2 - 6x - 6$, $2x^3 / x^3 = 2$, $(2x^3 + 12x^2 - 6x - 6) - (2)(x^3+6x^2-3x-5) = 4$.
- The quotient is $-3x^2 - 4x + 2$ and the remainder is $4$, so the final answer is $\boxed{-3x^2 - 4x + 2 + \frac{4}{x^3+6x^2-3x-5}}$.
### Explanation
1. Understanding the Problem
We are asked to perform polynomial long division to divide $-3 x^5-22 x^4-13 x^3+39 x^2+14 x-6$ by $x^3+6 x^2-3 x-5$. This means we want to find polynomials $q(x)$ (the quotient) and $r(x)$ (the remainder) such that $-3 x^5-22 x^4-13 x^3+39 x^2+14 x-6 = (x^3+6 x^2-3 x-5)q(x) + r(x)$, where the degree of $r(x)$ is less than the degree of $x^3+6 x^2-3 x-5$, which is 3.
2. Finding the First Term of the Quotient
We start by dividing the highest degree term of the dividend, $-3x^5$, by the highest degree term of the divisor, $x^3$. This gives us $-3x^2$, which is the first term of the quotient.
3. Subtracting and Bringing Down
Next, we multiply the divisor, $x^3+6 x^2-3 x-5$, by $-3x^2$ to get $-3x^5 - 18x^4 + 9x^3 + 15x^2$. We subtract this from the dividend: $(-3 x^5-22 x^4-13 x^3+39 x^2+14 x-6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2) = -4x^4 - 22x^3 + 24x^2 + 14x - 6$.
4. Finding the Second Term of the Quotient
Now we repeat the process with the new dividend, $-4x^4 - 22x^3 + 24x^2 + 14x - 6$. We divide the highest degree term, $-4x^4$, by the highest degree term of the divisor, $x^3$, which gives us $-4x$. This is the next term of the quotient.
5. Subtracting Again
We multiply the divisor, $x^3+6 x^2-3 x-5$, by $-4x$ to get $-4x^4 - 24x^3 + 12x^2 + 20x$. We subtract this from the new dividend: $(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x) = 2x^3 + 12x^2 - 6x - 6$.
6. Finding the Third Term of the Quotient
We repeat the process again with the new dividend, $2x^3 + 12x^2 - 6x - 6$. We divide the highest degree term, $2x^3$, by the highest degree term of the divisor, $x^3$, which gives us $2$. This is the next term of the quotient.
7. Final Subtraction
We multiply the divisor, $x^3+6 x^2-3 x-5$, by $2$ to get $2x^3 + 12x^2 - 6x - 10$. We subtract this from the new dividend: $(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10) = 4$.
8. Determining the Quotient and Remainder
The remainder is $4$, which has a degree of 0, less than the degree of the divisor, which is 3. Therefore, the quotient is $-3x^2 - 4x + 2$ and the remainder is $4$.
9. Final Answer
Thus, $\frac{-3 x^5-22 x^4-13 x^3+39 x^2+14 x-6}{x^3+6 x^2-3 x-5} = -3x^2 - 4x + 2 + \frac{4}{x^3+6 x^2-3 x-5}$. The quotient is $-3x^2 - 4x + 2$ and the remainder is $4$.
10. Conclusion
The result of the long division is a quotient of $-3x^2 - 4x + 2$ and a remainder of $4$. Therefore, the answer is $-3x^2 - 4x + 2 + \frac{4}{x^3+6x^2-3x-5}$.
### Examples
Polynomial long division is used in various engineering and scientific applications. For instance, in control systems, it helps simplify transfer functions to analyze system stability and response. In signal processing, it can decompose complex signals into simpler components for easier analysis and manipulation. Moreover, in computer graphics, polynomial division can be used to optimize rendering algorithms by simplifying complex polynomial expressions that define curves and surfaces, leading to faster and more efficient graphics processing.
