Answer :
Let's solve each part of the given problem step by step.
### 1. Divide:
#### (i) [tex]\( 38x^3 - 57x^2 \)[/tex] by [tex]\( 19x \)[/tex]:
To divide [tex]\( 38x^3 - 57x^2 \)[/tex] by [tex]\( 19x \)[/tex], we need to divide each term separately:
[tex]\[ \frac{38x^3}{19x} = 2x^2 \][/tex]
[tex]\[ \frac{57x^2}{19x} = 3x \][/tex]
Therefore, the result of the division is:
[tex]\[ 2x^2 - 3x \][/tex]
So, [tex]\((38x^3 - 57x^2) \div 19x = 2x^2 - 3x\)[/tex].
#### (iii) [tex]\( 28a^3 - 21a^2 + 7aby \)[/tex] by [tex]\( 7a \)[/tex]:
To divide [tex]\( 28a^3 - 21a^2 + 7aby \)[/tex] by [tex]\( 7a \)[/tex], we divide each term separately:
[tex]\[ \frac{28a^3}{7a} = 4a^2 \][/tex]
[tex]\[ \frac{21a^2}{7a} = 3a \][/tex]
[tex]\[ \frac{7aby}{7a} = by \][/tex]
Therefore, the result of the division is:
[tex]\[ 4a^2 - 3a + by \][/tex]
So, [tex]\((28a^3 - 21a^2 + 7aby) \div 7a = by + 4a^2 - 3a\)[/tex].
#### (ii) [tex]\( 23x^4 + 69x^3 - 46x^2 \)[/tex] by [tex]\( -23x^2 \)[/tex]:
To divide [tex]\( 23x^4 + 69x^3 - 46x^2 \)[/tex] by [tex]\( -23x^2 \)[/tex], we divide each term:
[tex]\[ \frac{23x^4}{-23x^2} = -x^2 \][/tex]
[tex]\[ \frac{69x^3}{-23x^2} = -3x \][/tex]
[tex]\[ \frac{46x^2}{-23x^2} = -2 \][/tex]
Therefore, the result of the division is:
[tex]\[ -x^2 - 3x + 2 \][/tex]
So, [tex]\((23x^4 + 69x^3 - 46x^2) \div -23x^2 = -x^2 - 3x + 2\)[/tex].
#### (ii) [tex]\( 5a^2b^2 - 22a^2b^2 + 11a^2b^2 \)[/tex] by [tex]\( 11a^2x^2 \)[/tex]:
To divide [tex]\( 5a^2b^2 - 22a^2b^2 + 11a^2b^2 \)[/tex] by [tex]\( 11a^2x^2 \)[/tex]:
- Simplify the numerator: [tex]\( (5a^2b^2 - 22a^2b^2 + 11a^2b^2 = 6a^2b^2) \)[/tex]
- Divide by [tex]\( 11a^2x^2 \)[/tex]:
[tex]\[ \frac{6a^2b^2}{11a^2x^2} = \frac{6b^2}{11x^2} \][/tex]
However, since the denominator term doesn’t divide properly (as seen from the given context), the polynomial quotient is effectively:
[tex]\[ 0 \][/tex]
The remainder will therefore be:
[tex]\[ -6a^2b^2 \][/tex]
So, the division [tex]\( (5a^2b^2 - 22a^2b^2 + 11a^2b^2) \div 11a^2x^2 = 0 \text{ with a remainder } -6a^2b^2.\)[/tex]
#### (v) [tex]\( 2xy \geq z + 3xy^2z^2 - 4x^2yz \)[/tex] by [tex]\( xyz \)[/tex]:
To divide [tex]\( 2xy + 3xy^2z^2 - 4x^2yz \)[/tex] by [tex]\( xyz \)[/tex]:
[tex]\[ \frac{2xy}{xyz} = \frac{2}{z} \][/tex]
[tex]\[ \frac{3xy^2z^2}{xyz} = 3xy \][/tex]
[tex]\[ \frac{4x^2yz}{xyz} = 4x \][/tex]
Therefore, the division results in:
[tex]\[ 3xy + \frac{2}{z} - 4x \][/tex]
Thus, the partial result remains, but the division proceeds with:
So, [tex]\( 2xy \geq z + 3xy^2z^2 - 4x^2yz \div xyz = -4x \text{ with a quotient } 3xy + \frac{2}{z}. \)[/tex]
#### (ㅊ) [tex]\( 8x^4 - 16x^3 + 24x^{-1} + 8x^3 \)[/tex] by [tex]\( 8x^2 \)[/tex]:
To divide [tex]\( 8x^4 - 16x^3 + 24x^{-1} + 8x^3 \)[/tex] by [tex]\( 8x^2 \)[/tex]:
[tex]\[ \frac{8x^4}{8x^2} = x^2 \][/tex]
[tex]\[ \frac{16x^3}{8x^2} = 2x \][/tex]
[tex]\[ \frac{24x^{-1}}{8x^2} = 3/x \][/tex]
Thus, [tex]\( 8x^4 - 16x^3 + 24x^{-1} + 8x^3 \equiv x^2 - x + \frac{24}{x} \)[/tex].
So, [tex]\((8x^4 - 16x^3 + 24x^{-1} + 8x^3) \div 8x^2 = x^2 - x + \frac{24 \cdot 1}{x}.\)[/tex]
#### (vii) [tex]\( 12x^2y^2 - 6x^2y^2 + 18x^2y^2z^2 - 54x^2yz \)[/tex] by [tex]\( 6x^2yz \)[/tex]):
To divide [tex]\( 12x^2y^2 - 6x^2y^2 + 18x^2y^2z^2 - 54x^2yz \)[/tex] by [tex]\( 6x^2yz \)[/tex]:
[tex]\[ \frac{12x^2y^2}{6x^2yz} = 2x/z \][/tex]
[tex]\[ \frac{6x^2y^2}{6x^2yz} = y \equiv y(2x - yz) \][/tex]
As partial results show higher order terms:
[tex]\[ \text{Division results: }13x^2y^2z^2-54x^2 = quotient:18x^2y^2z^2\][/tex]
#### (viii) [tex]\( 3x^5 - 9x^4 - 6x^2 \)[/tex] by [tex]\( 3x^2 \)[/tex]:
To divide [tex]\( 3x^5 - 9x^4 - 6x^2 \)[/tex] by [tex]\( 3x^2 \)[/tex]:
[tex]\[ \frac{3x^5}{3x^2} = x^3 \][/tex]
[tex]\[ \frac{9x^4}{3x^2} = 3x^2 \][/tex]
[tex]\[ \frac{6x^2}{3x^2} = -2\][/tex]
Therefore, merging:
[tex]\[x^5-x^{-1}+2x\][/tex]
Thus, [tex]\[((3x^5 -9x^{-6} -6.x^x)) \div 3x^2)x^3 -3x -2\][/tex]
### 2. Divide:
[tex]\[\][/tex]
Thus, the complete analysis is supported.
### 1. Divide:
#### (i) [tex]\( 38x^3 - 57x^2 \)[/tex] by [tex]\( 19x \)[/tex]:
To divide [tex]\( 38x^3 - 57x^2 \)[/tex] by [tex]\( 19x \)[/tex], we need to divide each term separately:
[tex]\[ \frac{38x^3}{19x} = 2x^2 \][/tex]
[tex]\[ \frac{57x^2}{19x} = 3x \][/tex]
Therefore, the result of the division is:
[tex]\[ 2x^2 - 3x \][/tex]
So, [tex]\((38x^3 - 57x^2) \div 19x = 2x^2 - 3x\)[/tex].
#### (iii) [tex]\( 28a^3 - 21a^2 + 7aby \)[/tex] by [tex]\( 7a \)[/tex]:
To divide [tex]\( 28a^3 - 21a^2 + 7aby \)[/tex] by [tex]\( 7a \)[/tex], we divide each term separately:
[tex]\[ \frac{28a^3}{7a} = 4a^2 \][/tex]
[tex]\[ \frac{21a^2}{7a} = 3a \][/tex]
[tex]\[ \frac{7aby}{7a} = by \][/tex]
Therefore, the result of the division is:
[tex]\[ 4a^2 - 3a + by \][/tex]
So, [tex]\((28a^3 - 21a^2 + 7aby) \div 7a = by + 4a^2 - 3a\)[/tex].
#### (ii) [tex]\( 23x^4 + 69x^3 - 46x^2 \)[/tex] by [tex]\( -23x^2 \)[/tex]:
To divide [tex]\( 23x^4 + 69x^3 - 46x^2 \)[/tex] by [tex]\( -23x^2 \)[/tex], we divide each term:
[tex]\[ \frac{23x^4}{-23x^2} = -x^2 \][/tex]
[tex]\[ \frac{69x^3}{-23x^2} = -3x \][/tex]
[tex]\[ \frac{46x^2}{-23x^2} = -2 \][/tex]
Therefore, the result of the division is:
[tex]\[ -x^2 - 3x + 2 \][/tex]
So, [tex]\((23x^4 + 69x^3 - 46x^2) \div -23x^2 = -x^2 - 3x + 2\)[/tex].
#### (ii) [tex]\( 5a^2b^2 - 22a^2b^2 + 11a^2b^2 \)[/tex] by [tex]\( 11a^2x^2 \)[/tex]:
To divide [tex]\( 5a^2b^2 - 22a^2b^2 + 11a^2b^2 \)[/tex] by [tex]\( 11a^2x^2 \)[/tex]:
- Simplify the numerator: [tex]\( (5a^2b^2 - 22a^2b^2 + 11a^2b^2 = 6a^2b^2) \)[/tex]
- Divide by [tex]\( 11a^2x^2 \)[/tex]:
[tex]\[ \frac{6a^2b^2}{11a^2x^2} = \frac{6b^2}{11x^2} \][/tex]
However, since the denominator term doesn’t divide properly (as seen from the given context), the polynomial quotient is effectively:
[tex]\[ 0 \][/tex]
The remainder will therefore be:
[tex]\[ -6a^2b^2 \][/tex]
So, the division [tex]\( (5a^2b^2 - 22a^2b^2 + 11a^2b^2) \div 11a^2x^2 = 0 \text{ with a remainder } -6a^2b^2.\)[/tex]
#### (v) [tex]\( 2xy \geq z + 3xy^2z^2 - 4x^2yz \)[/tex] by [tex]\( xyz \)[/tex]:
To divide [tex]\( 2xy + 3xy^2z^2 - 4x^2yz \)[/tex] by [tex]\( xyz \)[/tex]:
[tex]\[ \frac{2xy}{xyz} = \frac{2}{z} \][/tex]
[tex]\[ \frac{3xy^2z^2}{xyz} = 3xy \][/tex]
[tex]\[ \frac{4x^2yz}{xyz} = 4x \][/tex]
Therefore, the division results in:
[tex]\[ 3xy + \frac{2}{z} - 4x \][/tex]
Thus, the partial result remains, but the division proceeds with:
So, [tex]\( 2xy \geq z + 3xy^2z^2 - 4x^2yz \div xyz = -4x \text{ with a quotient } 3xy + \frac{2}{z}. \)[/tex]
#### (ㅊ) [tex]\( 8x^4 - 16x^3 + 24x^{-1} + 8x^3 \)[/tex] by [tex]\( 8x^2 \)[/tex]:
To divide [tex]\( 8x^4 - 16x^3 + 24x^{-1} + 8x^3 \)[/tex] by [tex]\( 8x^2 \)[/tex]:
[tex]\[ \frac{8x^4}{8x^2} = x^2 \][/tex]
[tex]\[ \frac{16x^3}{8x^2} = 2x \][/tex]
[tex]\[ \frac{24x^{-1}}{8x^2} = 3/x \][/tex]
Thus, [tex]\( 8x^4 - 16x^3 + 24x^{-1} + 8x^3 \equiv x^2 - x + \frac{24}{x} \)[/tex].
So, [tex]\((8x^4 - 16x^3 + 24x^{-1} + 8x^3) \div 8x^2 = x^2 - x + \frac{24 \cdot 1}{x}.\)[/tex]
#### (vii) [tex]\( 12x^2y^2 - 6x^2y^2 + 18x^2y^2z^2 - 54x^2yz \)[/tex] by [tex]\( 6x^2yz \)[/tex]):
To divide [tex]\( 12x^2y^2 - 6x^2y^2 + 18x^2y^2z^2 - 54x^2yz \)[/tex] by [tex]\( 6x^2yz \)[/tex]:
[tex]\[ \frac{12x^2y^2}{6x^2yz} = 2x/z \][/tex]
[tex]\[ \frac{6x^2y^2}{6x^2yz} = y \equiv y(2x - yz) \][/tex]
As partial results show higher order terms:
[tex]\[ \text{Division results: }13x^2y^2z^2-54x^2 = quotient:18x^2y^2z^2\][/tex]
#### (viii) [tex]\( 3x^5 - 9x^4 - 6x^2 \)[/tex] by [tex]\( 3x^2 \)[/tex]:
To divide [tex]\( 3x^5 - 9x^4 - 6x^2 \)[/tex] by [tex]\( 3x^2 \)[/tex]:
[tex]\[ \frac{3x^5}{3x^2} = x^3 \][/tex]
[tex]\[ \frac{9x^4}{3x^2} = 3x^2 \][/tex]
[tex]\[ \frac{6x^2}{3x^2} = -2\][/tex]
Therefore, merging:
[tex]\[x^5-x^{-1}+2x\][/tex]
Thus, [tex]\[((3x^5 -9x^{-6} -6.x^x)) \div 3x^2)x^3 -3x -2\][/tex]
### 2. Divide:
[tex]\[\][/tex]
Thus, the complete analysis is supported.
