Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To solve the problem of dividing the polynomial [tex]\( \frac{x^4 - 2x^3 - 195x^2}{x^3 - 3x^2 - 8x - 8} \)[/tex], we need to perform polynomial long division. Here is the step-by-step solution:
1. Setup the Division: We are dividing [tex]\( x^4 - 2x^3 - 195x^2 \)[/tex] by [tex]\( x^3 - 3x^2 - 8x - 8 \)[/tex].
2. First Division: Divide the leading term of the numerator ([tex]\( x^4 \)[/tex]) by the leading term of the denominator ([tex]\( x^3 \)[/tex]).
[tex]\[ \frac{x^4}{x^3} = x \][/tex]
Therefore, the first term of the quotient is [tex]\( x \)[/tex].
3. Multiply and Subtract: Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3x^2 - 8x - 8 \)[/tex] and subtract the result from the numerator:
[tex]\[ x \cdot (x^3 - 3x^2 - 8x - 8) = x^4 - 3x^3 - 8x^2 - 8x \][/tex]
[tex]\[ (x^4 - 2x^3 - 195x^2) - (x^4 - 3x^3 - 8x^2 - 8x) = x^4 - 2x^3 - 195x^2 - x^4 + 3x^3 + 8x^2 + 8x = x^3 - 187x^2 + 8x \][/tex]
4. Repeat the Process: The new numerator is [tex]\( x^3 - 187x^2 + 8x \)[/tex]. Now divide the leading term [tex]\( x^3 \)[/tex] by the leading term [tex]\( x^3 \)[/tex]:
[tex]\[ \frac{x^3}{x^3} = 1 \][/tex]
Therefore, the next term of the quotient is [tex]\( 1 \)[/tex].
5. Multiply and Subtract Again:
[tex]\[ 1 \cdot (x^3 - 3x^2 - 8x - 8) = x^3 - 3x^2 - 8x - 8 \][/tex]
[tex]\[ (x^3 - 187x^2 + 8x) - (x^3 - 3x^2 - 8x - 8) = x^3 - 187x^2 + 8x - x^3 + 3x^2 + 8x + 8 = -184x^2 + 16x + 8 \][/tex]
6. Remainder: Now the new numerator is [tex]\( -184x^2 + 16x + 8 \)[/tex], which has a degree lower than the degree of the divisor [tex]\( x^3 - 3x^2 - 8x - 8 \)[/tex]. Therefore, we stop here. This remaining polynomial is the remainder of the division.
7. Conclusion: The quotient of our division is [tex]\( x + 1 \)[/tex] and the remainder is [tex]\( -184x^2 + 16x + 8 \)[/tex].
Thus, the solution can be written as:
[tex]\[ \frac{x^4 - 2x^3 - 195x^2}{x^3 - 3x^2 - 8x - 8} = (x + 1) + \frac{-184x^2 + 16x + 8}{x^3 - 3x^2 - 8x - 8} \][/tex]
1. Setup the Division: We are dividing [tex]\( x^4 - 2x^3 - 195x^2 \)[/tex] by [tex]\( x^3 - 3x^2 - 8x - 8 \)[/tex].
2. First Division: Divide the leading term of the numerator ([tex]\( x^4 \)[/tex]) by the leading term of the denominator ([tex]\( x^3 \)[/tex]).
[tex]\[ \frac{x^4}{x^3} = x \][/tex]
Therefore, the first term of the quotient is [tex]\( x \)[/tex].
3. Multiply and Subtract: Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3x^2 - 8x - 8 \)[/tex] and subtract the result from the numerator:
[tex]\[ x \cdot (x^3 - 3x^2 - 8x - 8) = x^4 - 3x^3 - 8x^2 - 8x \][/tex]
[tex]\[ (x^4 - 2x^3 - 195x^2) - (x^4 - 3x^3 - 8x^2 - 8x) = x^4 - 2x^3 - 195x^2 - x^4 + 3x^3 + 8x^2 + 8x = x^3 - 187x^2 + 8x \][/tex]
4. Repeat the Process: The new numerator is [tex]\( x^3 - 187x^2 + 8x \)[/tex]. Now divide the leading term [tex]\( x^3 \)[/tex] by the leading term [tex]\( x^3 \)[/tex]:
[tex]\[ \frac{x^3}{x^3} = 1 \][/tex]
Therefore, the next term of the quotient is [tex]\( 1 \)[/tex].
5. Multiply and Subtract Again:
[tex]\[ 1 \cdot (x^3 - 3x^2 - 8x - 8) = x^3 - 3x^2 - 8x - 8 \][/tex]
[tex]\[ (x^3 - 187x^2 + 8x) - (x^3 - 3x^2 - 8x - 8) = x^3 - 187x^2 + 8x - x^3 + 3x^2 + 8x + 8 = -184x^2 + 16x + 8 \][/tex]
6. Remainder: Now the new numerator is [tex]\( -184x^2 + 16x + 8 \)[/tex], which has a degree lower than the degree of the divisor [tex]\( x^3 - 3x^2 - 8x - 8 \)[/tex]. Therefore, we stop here. This remaining polynomial is the remainder of the division.
7. Conclusion: The quotient of our division is [tex]\( x + 1 \)[/tex] and the remainder is [tex]\( -184x^2 + 16x + 8 \)[/tex].
Thus, the solution can be written as:
[tex]\[ \frac{x^4 - 2x^3 - 195x^2}{x^3 - 3x^2 - 8x - 8} = (x + 1) + \frac{-184x^2 + 16x + 8}{x^3 - 3x^2 - 8x - 8} \][/tex]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
