Answer :
To solve the problem of dividing the polynomial [tex]\(6x^4 - 7x^3 - 23x^2 + 18x + 20\)[/tex] by [tex]\(-3x + 5\)[/tex], we can use polynomial long division. Here's how you go about it:
1. Setup the Division:
- Write [tex]\(6x^4 - 7x^3 - 23x^2 + 18x + 20\)[/tex] inside the long division symbol.
- [tex]\(-3x + 5\)[/tex] is the divisor.
2. First Division Step:
- Divide the first term of the numerator [tex]\(6x^4\)[/tex] by the first term of the divisor [tex]\(-3x\)[/tex], which gives [tex]\(-2x^3\)[/tex].
- Multiply the entire divisor [tex]\(-3x + 5\)[/tex] by [tex]\(-2x^3\)[/tex] and subtract it from the current polynomial:
[tex]\((-3x + 5) \times (-2x^3) = 6x^4 - 10x^3\)[/tex].
- Subtract: [tex]\((6x^4 - 7x^3) - (6x^4 - 10x^3) = 3x^3\)[/tex].
3. Second Division Step:
- Bring down the next term [tex]\(-23x^2\)[/tex] to work with [tex]\(3x^3\)[/tex].
- Divide [tex]\(3x^3\)[/tex] by [tex]\(-3x\)[/tex], which gives [tex]\(-x^2\)[/tex].
- Multiply the divisor [tex]\(-3x + 5\)[/tex] by [tex]\(-x^2\)[/tex] and subtract:
[tex]\((-3x + 5) \times (-x^2) = 3x^3 - 5x^2\)[/tex].
- Subtract: [tex]\((3x^3 - 23x^2) - (3x^3 - 5x^2) = -18x^2\)[/tex].
4. Third Division Step:
- Bring down the next term [tex]\(+18x\)[/tex] to get [tex]\(-18x^2 + 18x\)[/tex].
- Divide [tex]\(-18x^2\)[/tex] by [tex]\(-3x\)[/tex], which gives [tex]\(6x\)[/tex].
- Multiply the divisor [tex]\(-3x + 5\)[/tex] by [tex]\(6x\)[/tex] and subtract:
[tex]\((-3x + 5) \times 6x = -18x^2 + 30x\)[/tex].
- Subtract: [tex]\((-18x^2 + 18x) - (-18x^2 + 30x) = -12x\)[/tex].
5. Fourth Division Step:
- Bring down the final term [tex]\(+20\)[/tex] to work with [tex]\(-12x\)[/tex].
- Divide [tex]\(-12x\)[/tex] by [tex]\(-3x\)[/tex], which gives [tex]\(4\)[/tex].
- Multiply the divisor [tex]\(-3x + 5\)[/tex] by [tex]\(4\)[/tex] and subtract:
[tex]\((-3x + 5) \times 4 = -12x + 20\)[/tex].
- Subtract: [tex]\((-12x + 20) - (-12x + 20) = 0\)[/tex].
6. Conclusion:
- The quotient of the division is [tex]\(-2x^3 - x^2 + 6x + 4\)[/tex], and the remainder is [tex]\(0\)[/tex].
Therefore, the polynomial [tex]\(6x^4 - 7x^3 - 23x^2 + 18x + 20\)[/tex] divided by [tex]\(-3x + 5\)[/tex] equals [tex]\(-2x^3 - x^2 + 6x + 4\)[/tex].
1. Setup the Division:
- Write [tex]\(6x^4 - 7x^3 - 23x^2 + 18x + 20\)[/tex] inside the long division symbol.
- [tex]\(-3x + 5\)[/tex] is the divisor.
2. First Division Step:
- Divide the first term of the numerator [tex]\(6x^4\)[/tex] by the first term of the divisor [tex]\(-3x\)[/tex], which gives [tex]\(-2x^3\)[/tex].
- Multiply the entire divisor [tex]\(-3x + 5\)[/tex] by [tex]\(-2x^3\)[/tex] and subtract it from the current polynomial:
[tex]\((-3x + 5) \times (-2x^3) = 6x^4 - 10x^3\)[/tex].
- Subtract: [tex]\((6x^4 - 7x^3) - (6x^4 - 10x^3) = 3x^3\)[/tex].
3. Second Division Step:
- Bring down the next term [tex]\(-23x^2\)[/tex] to work with [tex]\(3x^3\)[/tex].
- Divide [tex]\(3x^3\)[/tex] by [tex]\(-3x\)[/tex], which gives [tex]\(-x^2\)[/tex].
- Multiply the divisor [tex]\(-3x + 5\)[/tex] by [tex]\(-x^2\)[/tex] and subtract:
[tex]\((-3x + 5) \times (-x^2) = 3x^3 - 5x^2\)[/tex].
- Subtract: [tex]\((3x^3 - 23x^2) - (3x^3 - 5x^2) = -18x^2\)[/tex].
4. Third Division Step:
- Bring down the next term [tex]\(+18x\)[/tex] to get [tex]\(-18x^2 + 18x\)[/tex].
- Divide [tex]\(-18x^2\)[/tex] by [tex]\(-3x\)[/tex], which gives [tex]\(6x\)[/tex].
- Multiply the divisor [tex]\(-3x + 5\)[/tex] by [tex]\(6x\)[/tex] and subtract:
[tex]\((-3x + 5) \times 6x = -18x^2 + 30x\)[/tex].
- Subtract: [tex]\((-18x^2 + 18x) - (-18x^2 + 30x) = -12x\)[/tex].
5. Fourth Division Step:
- Bring down the final term [tex]\(+20\)[/tex] to work with [tex]\(-12x\)[/tex].
- Divide [tex]\(-12x\)[/tex] by [tex]\(-3x\)[/tex], which gives [tex]\(4\)[/tex].
- Multiply the divisor [tex]\(-3x + 5\)[/tex] by [tex]\(4\)[/tex] and subtract:
[tex]\((-3x + 5) \times 4 = -12x + 20\)[/tex].
- Subtract: [tex]\((-12x + 20) - (-12x + 20) = 0\)[/tex].
6. Conclusion:
- The quotient of the division is [tex]\(-2x^3 - x^2 + 6x + 4\)[/tex], and the remainder is [tex]\(0\)[/tex].
Therefore, the polynomial [tex]\(6x^4 - 7x^3 - 23x^2 + 18x + 20\)[/tex] divided by [tex]\(-3x + 5\)[/tex] equals [tex]\(-2x^3 - x^2 + 6x + 4\)[/tex].
