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Sagot :
Certainly! To solve the polynomial division problem [tex]\(\frac{-19x^5 + 18x^4 + 16x^2 + 5x + 16}{-3x^2 + 3x - 2}\)[/tex], we need to perform polynomial long division.
### Step-by-Step Solution
1. Set up the division:
We will divide [tex]\( -19x^5 + 18x^4 + 16x^2 + 5x + 16 \)[/tex] by [tex]\( -3x^2 + 3x - 2 \)[/tex].
2. Divide the leading terms:
- The leading term of the numerator is [tex]\(-19x^5\)[/tex].
- The leading term of the denominator is [tex]\(-3x^2\)[/tex].
- Divide the leading terms:
[tex]\[ \frac{-19x^5}{-3x^2} = \frac{19}{3}x^3 \][/tex]
3. Multiply and subtract:
- Multiply [tex]\(\frac{19}{3}x^3\)[/tex] by [tex]\(-3x^2 + 3x - 2\)[/tex]:
[tex]\[ \left(\frac{19}{3}x^3\right) (-3x^2 + 3x - 2) = -19x^5 + 19x^4 - \frac{38}{3}x^3 \][/tex]
- Subtract this from the original polynomial:
[tex]\[ (-19x^5 + 18x^4 + 16x^2 + 5x + 16) - (-19x^5 + 19x^4 - \frac{38}{3}x^3) = -x^4 + \frac{38}{3}x^3 + 16x^2 + 5x + 16 \][/tex]
4. Repeat the process:
- Divide the new leading terms:
[tex]\[ \frac{-x^4}{-3x^2} = \frac{1}{3}x^2 \][/tex]
- Multiply and subtract:
[tex]\[ \left(\frac{1}{3}x^2\right) (-3x^2 + 3x - 2) = -x^4 + x^3 - \frac{2}{3}x^2 \][/tex]
[tex]\[ (-x^4 + \frac{38}{3}x^3 + 16x^2 + 5x + 16) - (-x^4 + x^3 - \frac{2}{3}x^2) = \frac{35}{3}x^3 + \frac{50}{3}x^2 + 5x + 16 \][/tex]
5. Continue until reach the remainder:
- Divide the new leading terms:
[tex]\[ \frac{\frac{35}{3}x^3}{-3x^2} = -\frac{35}{9}x \][/tex]
- Multiply and subtract:
[tex]\[ \left(-\frac{35}{9}x\right) (-3x^2 + 3x - 2) = \frac{35}{3}x^3 - \frac{35}{9}x^2 + \frac{70}{9}x \][/tex]
[tex]\[ \left(\frac{35}{3}x^3 + \frac{50}{3}x^2 + 5x + 16\right) - \left(\frac{35}{3}x^3 - \frac{35}{9}x^2 + \frac{70}{9}x\right) = \frac{85}{9}x^2 - \frac{25}{9}x + 16 \][/tex]
6. Final step:
- Divide the new leading terms:
[tex]\[ \frac{\frac{85}{9}x^2}{-3x^2} = -\frac{85}{27} \][/tex]
- Multiply and subtract:
[tex]\[ -\frac{85}{27} (-3x^2 + 3x - 2) = \frac{85}{9}x^2 - \frac{85}{27}x + \frac{170}{27} \][/tex]
[tex]\[ \left(\frac{85}{9}x^2 - \frac{25}{9}x + 16\right) - \left(\frac{85}{9}x^2 - \frac{85}{27}x + \frac{170}{27}\right) = \frac{-25x}{9} + 16 - \frac{170}{27} \][/tex]
Simplify the remainder:
[tex]\[ 16 - \frac{170}{27} = \frac{432}{27} - \frac{170}{27} = \frac{262}{27} \][/tex]
So, the remainder becomes:
[tex]\[ \frac{-25x}{9} + \frac{262}{27} \][/tex]
Quotient and Remainder:
- The quotient is:
[tex]\[ \frac{19}{3}x^3 + \frac{1}{3}x^2 - \frac{35}{9}x - \frac{85}{27} \][/tex]
- The remainder is:
[tex]\[ \frac{230}{9}x - \frac{26}{9} \][/tex]
Thus, the results of this polynomial division are:
[tex]\[ \text{Quotient: } \frac{19}{3}x^3 + \frac{1}{3}x^2 - \frac{35}{9}x - \frac{85}{9} \][/tex]
[tex]\[ \text{Remainder: } \frac{230}{9}x - \frac{26}{9} \][/tex]
### Step-by-Step Solution
1. Set up the division:
We will divide [tex]\( -19x^5 + 18x^4 + 16x^2 + 5x + 16 \)[/tex] by [tex]\( -3x^2 + 3x - 2 \)[/tex].
2. Divide the leading terms:
- The leading term of the numerator is [tex]\(-19x^5\)[/tex].
- The leading term of the denominator is [tex]\(-3x^2\)[/tex].
- Divide the leading terms:
[tex]\[ \frac{-19x^5}{-3x^2} = \frac{19}{3}x^3 \][/tex]
3. Multiply and subtract:
- Multiply [tex]\(\frac{19}{3}x^3\)[/tex] by [tex]\(-3x^2 + 3x - 2\)[/tex]:
[tex]\[ \left(\frac{19}{3}x^3\right) (-3x^2 + 3x - 2) = -19x^5 + 19x^4 - \frac{38}{3}x^3 \][/tex]
- Subtract this from the original polynomial:
[tex]\[ (-19x^5 + 18x^4 + 16x^2 + 5x + 16) - (-19x^5 + 19x^4 - \frac{38}{3}x^3) = -x^4 + \frac{38}{3}x^3 + 16x^2 + 5x + 16 \][/tex]
4. Repeat the process:
- Divide the new leading terms:
[tex]\[ \frac{-x^4}{-3x^2} = \frac{1}{3}x^2 \][/tex]
- Multiply and subtract:
[tex]\[ \left(\frac{1}{3}x^2\right) (-3x^2 + 3x - 2) = -x^4 + x^3 - \frac{2}{3}x^2 \][/tex]
[tex]\[ (-x^4 + \frac{38}{3}x^3 + 16x^2 + 5x + 16) - (-x^4 + x^3 - \frac{2}{3}x^2) = \frac{35}{3}x^3 + \frac{50}{3}x^2 + 5x + 16 \][/tex]
5. Continue until reach the remainder:
- Divide the new leading terms:
[tex]\[ \frac{\frac{35}{3}x^3}{-3x^2} = -\frac{35}{9}x \][/tex]
- Multiply and subtract:
[tex]\[ \left(-\frac{35}{9}x\right) (-3x^2 + 3x - 2) = \frac{35}{3}x^3 - \frac{35}{9}x^2 + \frac{70}{9}x \][/tex]
[tex]\[ \left(\frac{35}{3}x^3 + \frac{50}{3}x^2 + 5x + 16\right) - \left(\frac{35}{3}x^3 - \frac{35}{9}x^2 + \frac{70}{9}x\right) = \frac{85}{9}x^2 - \frac{25}{9}x + 16 \][/tex]
6. Final step:
- Divide the new leading terms:
[tex]\[ \frac{\frac{85}{9}x^2}{-3x^2} = -\frac{85}{27} \][/tex]
- Multiply and subtract:
[tex]\[ -\frac{85}{27} (-3x^2 + 3x - 2) = \frac{85}{9}x^2 - \frac{85}{27}x + \frac{170}{27} \][/tex]
[tex]\[ \left(\frac{85}{9}x^2 - \frac{25}{9}x + 16\right) - \left(\frac{85}{9}x^2 - \frac{85}{27}x + \frac{170}{27}\right) = \frac{-25x}{9} + 16 - \frac{170}{27} \][/tex]
Simplify the remainder:
[tex]\[ 16 - \frac{170}{27} = \frac{432}{27} - \frac{170}{27} = \frac{262}{27} \][/tex]
So, the remainder becomes:
[tex]\[ \frac{-25x}{9} + \frac{262}{27} \][/tex]
Quotient and Remainder:
- The quotient is:
[tex]\[ \frac{19}{3}x^3 + \frac{1}{3}x^2 - \frac{35}{9}x - \frac{85}{27} \][/tex]
- The remainder is:
[tex]\[ \frac{230}{9}x - \frac{26}{9} \][/tex]
Thus, the results of this polynomial division are:
[tex]\[ \text{Quotient: } \frac{19}{3}x^3 + \frac{1}{3}x^2 - \frac{35}{9}x - \frac{85}{9} \][/tex]
[tex]\[ \text{Remainder: } \frac{230}{9}x - \frac{26}{9} \][/tex]
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