Answer :
Let's divide each polynomial as specified in the exercise and show the step-by-step solutions:
1. Divide:
(i) [tex]\( 38x^3 - 57x^2 \)[/tex] by [tex]\( 19x \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{38x^3 - 57x^2}{19x} = \frac{38x^3}{19x} - \frac{57x^2}{19x} = 2x^2 - 3x \][/tex]
Simplifying the coefficients, we get:
[tex]\[ 2x^2 - 3x \][/tex]
Numerical coefficient:
[tex]\[ 38 - \frac{57}{19} = 35.0 \][/tex]
2. Divide:
(ii) [tex]\( 23x^4 + 69x^3 - 46x^2 \)[/tex] by [tex]\( -23x^2 \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{23x^4 + 69x^3 - 46x^2}{-23x^2} = \frac{23x^4}{-23x^2} + \frac{69x^3}{-23x^2} - \frac{46x^2}{-23x^2} = -x^2 - 3x + 2 \][/tex]
Numerical coefficients:
[tex]\[ (-23, -\frac{69}{23}, \frac{46}{-23}) = (-23, -3.0, -2.0) \][/tex]
3. Divide:
(iii) [tex]\( 28a^3 - 21a^2 + 7a \)[/tex] by [tex]\( 9a \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{28a^3 - 21a^2 + 7a}{9a} = \frac{28a^3}{9a} - \frac{21a^2}{9a} + \frac{7a}{9a} = \frac{28}{9}a^2 - \frac{21}{9}a + \frac{7}{9} \][/tex]
Simplifying the coefficients:
[tex]\[ \frac{28}{9} - \frac{21}{9} + \frac{7}{9} = 3.1111111111 - 2.3333333333 + 0.7777777777 \approx 1.5555555555555554 \][/tex]
4. Divide:
(iiv) [tex]\( 55a^4b^5 - 22a^2b^5 + 11a^2b^2 \)[/tex] by [tex]\( 11a^2b^2 \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{55a^4b^5 - 22a^2b^5 + 11a^2b^2}{11a^2b^2} = \frac{55a^4b^5}{11a^2b^2} - \frac{22a^2b^5}{11a^2b^2} + \frac{11a^2b^2}{11a^2b^2} = 5a^2b^3 - 2b^3 + 1 \][/tex]
Simplifying the coefficients:
[tex]\[ (5.0, -2.0, 1.0) \][/tex]
5. Divide:
(v) [tex]\( 2x^2y^2z + 3xy^2z^2 - 4x^2yz \)[/tex] by [tex]\( xyz \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{2x^2y^2z + 3xy^2z^2 - 4x^2yz}{xyz} = \frac{2x^2y^2z}{xyz} + \frac{3xy^2z^2}{xyz} - \frac{4x^2yz}{xyz} = 2xy + 3yz - 4xz \][/tex]
Simplifying the coefficients:
[tex]\[ (2.0, 3.0, -4.0) \][/tex]
6. Divide:
(iii) [tex]\( 8x^6 - 16x^5 + 24x^4 + 8x^3 \)[/tex] by [tex]\( 8x^2 \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{8x^6 - 16x^5 + 24x^4 + 8x^3}{8x^2} = \frac{8x^6}{8x^2} - \frac{16x^5}{8x^2} + \frac{24x^4}{8x^2} + \frac{8x^3}{8x^2} = x^4 - 2x^3 + 3x^2 + x \][/tex]
Simplifying the coefficients:
[tex]\[ (1.0, -2.0, 3.0) \][/tex]
7. Divide:
(vii) [tex]\( 12x^3y^2z^3 - 6x^3y^3z^2 + 18x^4y^2z^2 - 54x^2yz \)[/tex] by [tex]\( 6x^2yz \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{12x^3y^2z^3 - 6x^3y^3z^2 + 18x^4y^2z^2 - 54x^2yz}{6x^2yz} = \frac{12x^3y^2z^3}{6x^2yz} - \frac{6x^3y^3z^2}{6x^2yz} + \frac{18x^4y^2z^2}{6x^2yz} - \frac{54x^2yz}{6x^2yz} = 2xy - y^2z + 3x^2yz - 9 \][/tex]
Simplifying the coefficients:
[tex]\[ (2.0, -1.0, 3.0, -9.0) \][/tex]
8. Divide:
(viii) [tex]\( 3x^5 - 9x^4 - 6x^2 \)[/tex] by [tex]\( 3x^2 \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{3x^5 - 9x^4 - 6x^2}{3x^2} = \frac{3x^5}{3x^2} - \frac{9x^4}{3x^2} - \frac{6x^2}{3x^2} = x^3 - 3x^2 - 2 \][/tex]
Simplifying the coefficients:
[tex]\[ (1.0, -3.0, -2.0) \][/tex]
1. Divide:
(i) [tex]\( 38x^3 - 57x^2 \)[/tex] by [tex]\( 19x \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{38x^3 - 57x^2}{19x} = \frac{38x^3}{19x} - \frac{57x^2}{19x} = 2x^2 - 3x \][/tex]
Simplifying the coefficients, we get:
[tex]\[ 2x^2 - 3x \][/tex]
Numerical coefficient:
[tex]\[ 38 - \frac{57}{19} = 35.0 \][/tex]
2. Divide:
(ii) [tex]\( 23x^4 + 69x^3 - 46x^2 \)[/tex] by [tex]\( -23x^2 \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{23x^4 + 69x^3 - 46x^2}{-23x^2} = \frac{23x^4}{-23x^2} + \frac{69x^3}{-23x^2} - \frac{46x^2}{-23x^2} = -x^2 - 3x + 2 \][/tex]
Numerical coefficients:
[tex]\[ (-23, -\frac{69}{23}, \frac{46}{-23}) = (-23, -3.0, -2.0) \][/tex]
3. Divide:
(iii) [tex]\( 28a^3 - 21a^2 + 7a \)[/tex] by [tex]\( 9a \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{28a^3 - 21a^2 + 7a}{9a} = \frac{28a^3}{9a} - \frac{21a^2}{9a} + \frac{7a}{9a} = \frac{28}{9}a^2 - \frac{21}{9}a + \frac{7}{9} \][/tex]
Simplifying the coefficients:
[tex]\[ \frac{28}{9} - \frac{21}{9} + \frac{7}{9} = 3.1111111111 - 2.3333333333 + 0.7777777777 \approx 1.5555555555555554 \][/tex]
4. Divide:
(iiv) [tex]\( 55a^4b^5 - 22a^2b^5 + 11a^2b^2 \)[/tex] by [tex]\( 11a^2b^2 \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{55a^4b^5 - 22a^2b^5 + 11a^2b^2}{11a^2b^2} = \frac{55a^4b^5}{11a^2b^2} - \frac{22a^2b^5}{11a^2b^2} + \frac{11a^2b^2}{11a^2b^2} = 5a^2b^3 - 2b^3 + 1 \][/tex]
Simplifying the coefficients:
[tex]\[ (5.0, -2.0, 1.0) \][/tex]
5. Divide:
(v) [tex]\( 2x^2y^2z + 3xy^2z^2 - 4x^2yz \)[/tex] by [tex]\( xyz \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{2x^2y^2z + 3xy^2z^2 - 4x^2yz}{xyz} = \frac{2x^2y^2z}{xyz} + \frac{3xy^2z^2}{xyz} - \frac{4x^2yz}{xyz} = 2xy + 3yz - 4xz \][/tex]
Simplifying the coefficients:
[tex]\[ (2.0, 3.0, -4.0) \][/tex]
6. Divide:
(iii) [tex]\( 8x^6 - 16x^5 + 24x^4 + 8x^3 \)[/tex] by [tex]\( 8x^2 \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{8x^6 - 16x^5 + 24x^4 + 8x^3}{8x^2} = \frac{8x^6}{8x^2} - \frac{16x^5}{8x^2} + \frac{24x^4}{8x^2} + \frac{8x^3}{8x^2} = x^4 - 2x^3 + 3x^2 + x \][/tex]
Simplifying the coefficients:
[tex]\[ (1.0, -2.0, 3.0) \][/tex]
7. Divide:
(vii) [tex]\( 12x^3y^2z^3 - 6x^3y^3z^2 + 18x^4y^2z^2 - 54x^2yz \)[/tex] by [tex]\( 6x^2yz \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{12x^3y^2z^3 - 6x^3y^3z^2 + 18x^4y^2z^2 - 54x^2yz}{6x^2yz} = \frac{12x^3y^2z^3}{6x^2yz} - \frac{6x^3y^3z^2}{6x^2yz} + \frac{18x^4y^2z^2}{6x^2yz} - \frac{54x^2yz}{6x^2yz} = 2xy - y^2z + 3x^2yz - 9 \][/tex]
Simplifying the coefficients:
[tex]\[ (2.0, -1.0, 3.0, -9.0) \][/tex]
8. Divide:
(viii) [tex]\( 3x^5 - 9x^4 - 6x^2 \)[/tex] by [tex]\( 3x^2 \)[/tex]
Step-by-Step Solution:
[tex]\[ \frac{3x^5 - 9x^4 - 6x^2}{3x^2} = \frac{3x^5}{3x^2} - \frac{9x^4}{3x^2} - \frac{6x^2}{3x^2} = x^3 - 3x^2 - 2 \][/tex]
Simplifying the coefficients:
[tex]\[ (1.0, -3.0, -2.0) \][/tex]
