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Sagot :
Certainly! Let's perform polynomial long division to divide [tex]\( \frac{6x^4 + 7x^3 - 9x^2 + 13x - 12}{3x^2 - x + 3} \)[/tex].
### Step-by-Step Division:
1. Divide the leading terms:
- Divide the leading term of the numerator [tex]\(6x^4\)[/tex] by the leading term of the denominator [tex]\(3x^2\)[/tex]:
[tex]\[ \frac{6x^4}{3x^2} = 2x^2 \][/tex]
2. Multiply the entire divisor by the obtained term:
- Multiply [tex]\(2x^2\)[/tex] by [tex]\(3x^2 - x + 3\)[/tex]:
[tex]\[ 2x^2 \cdot (3x^2 - x + 3) = 6x^4 - 2x^3 + 6x^2 \][/tex]
3. Subtract this from the original polynomial:
- Subtract [tex]\(6x^4 - 2x^3 + 6x^2\)[/tex] from [tex]\(6x^4 + 7x^3 - 9x^2 + 13x - 12\)[/tex]:
[tex]\[ (6x^4 + 7x^3 - 9x^2 + 13x - 12) - (6x^4 - 2x^3 + 6x^2) = (7x^3 + 2x^3 - 9x^2 - 6x^2 + 13x - 12) = 9x^3 - 15x^2 + 13x - 12 \][/tex]
4. Repeat the process with the new polynomial:
- Divide the leading term of the new polynomial [tex]\(9x^3\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[ \frac{9x^3}{3x^2} = 3x \][/tex]
- Multiply [tex]\(3x\)[/tex] by [tex]\(3x^2 - x + 3\)[/tex]:
[tex]\[ 3x \cdot (3x^2 - x + 3) = 9x^3 - 3x^2 + 9x \][/tex]
- Subtract:
[tex]\[ (9x^3 - 15x^2 + 13x - 12) - (9x^3 - 3x^2 + 9x) = (-15x^2 + 3x^2 + 13x - 9x - 12) = -12x^2 + 4x - 12 \][/tex]
5. Continue with the final quotient term:
- Divide the leading term [tex]\((-12x^2)\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[ \frac{-12x^2}{3x^2} = -4 \][/tex]
- Multiply [tex]\(-4\)[/tex] by [tex]\(3x^2 - x + 3\)[/tex]:
[tex]\[ -4 \cdot (3x^2 - x + 3) = -12x^2 + 4x - 12 \][/tex]
- Subtract:
[tex]\[ (-12x^2 + 4x - 12) - (-12x^2 + 4x - 12) = 0 \][/tex]
### Conclusion:
After completing the division, the quotient is [tex]\(2x^2 + 3x - 4\)[/tex] and the remainder is [tex]\(0\)[/tex].
So, the final answer is:
[tex]\[ \frac{6x^4 + 7x^3 - 9x^2 + 13x - 12}{3x^2 - x + 3} = 2x^2 + 3x - 4 \quad \text{with a remainder of } 0 \][/tex]
### Step-by-Step Division:
1. Divide the leading terms:
- Divide the leading term of the numerator [tex]\(6x^4\)[/tex] by the leading term of the denominator [tex]\(3x^2\)[/tex]:
[tex]\[ \frac{6x^4}{3x^2} = 2x^2 \][/tex]
2. Multiply the entire divisor by the obtained term:
- Multiply [tex]\(2x^2\)[/tex] by [tex]\(3x^2 - x + 3\)[/tex]:
[tex]\[ 2x^2 \cdot (3x^2 - x + 3) = 6x^4 - 2x^3 + 6x^2 \][/tex]
3. Subtract this from the original polynomial:
- Subtract [tex]\(6x^4 - 2x^3 + 6x^2\)[/tex] from [tex]\(6x^4 + 7x^3 - 9x^2 + 13x - 12\)[/tex]:
[tex]\[ (6x^4 + 7x^3 - 9x^2 + 13x - 12) - (6x^4 - 2x^3 + 6x^2) = (7x^3 + 2x^3 - 9x^2 - 6x^2 + 13x - 12) = 9x^3 - 15x^2 + 13x - 12 \][/tex]
4. Repeat the process with the new polynomial:
- Divide the leading term of the new polynomial [tex]\(9x^3\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[ \frac{9x^3}{3x^2} = 3x \][/tex]
- Multiply [tex]\(3x\)[/tex] by [tex]\(3x^2 - x + 3\)[/tex]:
[tex]\[ 3x \cdot (3x^2 - x + 3) = 9x^3 - 3x^2 + 9x \][/tex]
- Subtract:
[tex]\[ (9x^3 - 15x^2 + 13x - 12) - (9x^3 - 3x^2 + 9x) = (-15x^2 + 3x^2 + 13x - 9x - 12) = -12x^2 + 4x - 12 \][/tex]
5. Continue with the final quotient term:
- Divide the leading term [tex]\((-12x^2)\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[ \frac{-12x^2}{3x^2} = -4 \][/tex]
- Multiply [tex]\(-4\)[/tex] by [tex]\(3x^2 - x + 3\)[/tex]:
[tex]\[ -4 \cdot (3x^2 - x + 3) = -12x^2 + 4x - 12 \][/tex]
- Subtract:
[tex]\[ (-12x^2 + 4x - 12) - (-12x^2 + 4x - 12) = 0 \][/tex]
### Conclusion:
After completing the division, the quotient is [tex]\(2x^2 + 3x - 4\)[/tex] and the remainder is [tex]\(0\)[/tex].
So, the final answer is:
[tex]\[ \frac{6x^4 + 7x^3 - 9x^2 + 13x - 12}{3x^2 - x + 3} = 2x^2 + 3x - 4 \quad \text{with a remainder of } 0 \][/tex]
Answer:
2x² + 3x - 4
Step-by-step explanation:
To simplify the expression[tex]\dfrac{6x^4 + 7x^3 - 9x^2 + 13x - 12}{3x^2 - x + 3}[/tex], we will use polynomial long division.
1. Set up the division:
Dividend: [tex]6x^4 + 7x^3 - 9x^2 + 13x - 12[/tex]
Divisor:[tex]3x^2 - x + 3[/tex]
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\dfrac{6x^4}{3x^2} = 2x^2[/tex]
3. Multiply the entire divisor by 2x^2 and subtract from the original polynomial:
[tex]2x^2 \cdot (3x^2 - x + 3) = 6x^4 - 2x^3 + 6x^2\\\\ (6x^4 + 7x^3 - 9x^2 + 13x - 12) - (6x^4 - 2x^3 + 6x^2) \\\\ 9x^3 - 15x^2 + 13x - 12[/tex]
4. Repeat the process:
[tex]\dfrac{9x^3}{3x^2} = 3x[/tex]
Multiply the divisor by 3x and subtract:
[tex]3x \cdot (3x^2 - x + 3) = 9x^3 - 3x^2 + 9x\\\\ (9x^3 - 15x^2 + 13x - 12) - (9x^3 - 3x^2 + 9x) \\\\-12x^2 + 4x - 12[/tex]
5. Continue the process:
[tex]\dfrac{-12x^2}{3x^2} = -4[/tex]
Multiply the divisor by -4 and subtract:
[tex]-4 \cdot (3x^2 - x + 3) = -12x^2 + 4x - 12\\\\ (-12x^2 + 4x - 12) - (-12x^2 + 4x - 12) = 0[/tex]
The division is complete, and the quotient is:
[tex]2x^2 + 3x - 4[/tex]
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