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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]
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