Answer :
Sure, let's go through the polynomial division step-by-step for each question:
35. Divide [tex]\( \frac{9x - 6}{3} \)[/tex]:
- Divide each term by 3:
[tex]\[ \frac{9x}{3} = 3x \][/tex]
[tex]\[ \frac{-6}{3} = -2 \][/tex]
- So, the result is [tex]\( 3x - 2 \)[/tex].
36. Divide [tex]\( \frac{4x - 7}{2} \)[/tex]:
- Divide each term by 2:
[tex]\[ \frac{4x}{2} = 2x \][/tex]
[tex]\[ \frac{-7}{2} = -\frac{7}{2} \][/tex]
- The result is [tex]\( 2x - \frac{7}{2} \)[/tex].
37. Divide [tex]\( \frac{x^2 - 3x + 5}{x} \)[/tex]:
- Divide each term by [tex]\( x \)[/tex]:
[tex]\[ \frac{x^2}{x} = x \][/tex]
[tex]\[ \frac{-3x}{x} = -3 \][/tex]
[tex]\[ \frac{5}{x} \text{ can't be simplified and remains } 5 .\][/tex]
- The result is [tex]\( x - 3 + \frac{5}{x} \)[/tex].
38. Divide [tex]\( \frac{5x^2 - 25x + 2}{-5x} \)[/tex]:
- Divide each term by [tex]\( -5x \)[/tex]:
[tex]\[ \frac{5x^2}{-5x} = -x \][/tex]
[tex]\[ \frac{-25x}{-5x} = 5 \][/tex]
[tex]\[ \frac{2}{-5x} \text{ gives a remainder }.\][/tex]
- The result is [tex]\( -x + 5 + \frac{2}{-5x} \)[/tex].
39. Divide [tex]\( \frac{4x^{10} - 5x^9 - 20x^4}{4x^2} \)[/tex]:
- Divide each term by [tex]\( 4x^2 \)[/tex]:
[tex]\[ \frac{4x^{10}}{4x^2} = x^8 \][/tex]
[tex]\[ \frac{-5x^9}{4x^2} = -\frac{5}{4}x^7 \][/tex]
[tex]\[ \frac{-20x^4}{4x^2} = -5x^2 \][/tex]
- The result is [tex]\( x^8 - \frac{5}{4}x^7 - 5x^2 \)[/tex].
40. Divide [tex]\( \left(-x^6 + x^5 + 7x^2 - 9\right) \div x^4 \)[/tex]:
- Divide each term by [tex]\( x^4 \)[/tex]:
[tex]\[ \frac{-x^6}{x^4} = -x^2 \][/tex]
[tex]\[ \frac{x^5}{x^4} = x \][/tex]
[tex]\[ \frac{7x^2}{x^4} = \frac{7}{x^2} \text{ which remains in the solution} \][/tex]
[tex]\[ \frac{-9}{x^4} = -\frac{9}{x^4} \][/tex]
- The result is [tex]\( -x^2 + x + \frac{7x^2 - 9}{x^4} \)[/tex].
41. Divide [tex]\( \left(x^2 + 2x + 6\right) \div x \)[/tex]:
- Divide each term by [tex]\( x \)[/tex]:
[tex]\[ \frac{x^2}{x} = x \][/tex]
[tex]\[ \frac{2x}{x} = 2 \][/tex]
[tex]\[ \frac{6}{x} \text{ remains } 6 \][/tex]
- The result is [tex]\( x + 2 + \frac{6}{x} \)[/tex].
42. Divide [tex]\( \left(3x^2 - 15x + 5\right) \div (-3x) \)[/tex]:
- Divide each term by [tex]\( -3x \)[/tex]:
[tex]\[ \frac{3x^2}{-3x} = -x \][/tex]
[tex]\[ \frac{-15x}{-3x} = 5 \][/tex]
[tex]\[ \frac{5}{-3x} \text{ remains as a remainder} \][/tex]
- The result is [tex]\( -x + 5 + \frac{5}{-3x} \)[/tex].
43. Divide [tex]\( \left(2x^{11} - 5x^7 - 10x^6\right) \div 2x^3 \)[/tex]:
- Divide each term by [tex]\( 2x^3 \)[/tex]:
[tex]\[ \frac{2x^{11}}{2x^3} = x^8 \][/tex]
[tex]\[ \frac{-5x^7}{2x^3} = -\frac{5}{2}x^4 \][/tex]
[tex]\[ \frac{-10x^6}{2x^3} = -5x^3 \][/tex]
- The result is [tex]\( x^8 - \frac{5}{2}x^4 - 5x^3 \)[/tex].
44. Divide [tex]\( \left(-2x^6 + 5x^5 + 9x^2 + 2\right) \div x^4 \)[/tex]:
- Divide each term by [tex]\( x^4 \)[/tex]:
[tex]\[ \frac{-2x^6}{x^4} = -2x^2 \][/tex]
[tex]\[ \frac{5x^5}{x^4} = 5x \][/tex]
[tex]\[ \frac{9x^2}{x^4} = \frac{9}{x^2} \][/tex]
[tex]\[ \frac{2}{x^4} = \frac{2}{x^4} \][/tex]
- The result is [tex]\( -2x^2 + 5x + \frac{9x^2 + 2}{x^4} \)[/tex].
These steps help understand how each polynomial division is carried out. If you need further clarification on any specific step, feel free to ask!
35. Divide [tex]\( \frac{9x - 6}{3} \)[/tex]:
- Divide each term by 3:
[tex]\[ \frac{9x}{3} = 3x \][/tex]
[tex]\[ \frac{-6}{3} = -2 \][/tex]
- So, the result is [tex]\( 3x - 2 \)[/tex].
36. Divide [tex]\( \frac{4x - 7}{2} \)[/tex]:
- Divide each term by 2:
[tex]\[ \frac{4x}{2} = 2x \][/tex]
[tex]\[ \frac{-7}{2} = -\frac{7}{2} \][/tex]
- The result is [tex]\( 2x - \frac{7}{2} \)[/tex].
37. Divide [tex]\( \frac{x^2 - 3x + 5}{x} \)[/tex]:
- Divide each term by [tex]\( x \)[/tex]:
[tex]\[ \frac{x^2}{x} = x \][/tex]
[tex]\[ \frac{-3x}{x} = -3 \][/tex]
[tex]\[ \frac{5}{x} \text{ can't be simplified and remains } 5 .\][/tex]
- The result is [tex]\( x - 3 + \frac{5}{x} \)[/tex].
38. Divide [tex]\( \frac{5x^2 - 25x + 2}{-5x} \)[/tex]:
- Divide each term by [tex]\( -5x \)[/tex]:
[tex]\[ \frac{5x^2}{-5x} = -x \][/tex]
[tex]\[ \frac{-25x}{-5x} = 5 \][/tex]
[tex]\[ \frac{2}{-5x} \text{ gives a remainder }.\][/tex]
- The result is [tex]\( -x + 5 + \frac{2}{-5x} \)[/tex].
39. Divide [tex]\( \frac{4x^{10} - 5x^9 - 20x^4}{4x^2} \)[/tex]:
- Divide each term by [tex]\( 4x^2 \)[/tex]:
[tex]\[ \frac{4x^{10}}{4x^2} = x^8 \][/tex]
[tex]\[ \frac{-5x^9}{4x^2} = -\frac{5}{4}x^7 \][/tex]
[tex]\[ \frac{-20x^4}{4x^2} = -5x^2 \][/tex]
- The result is [tex]\( x^8 - \frac{5}{4}x^7 - 5x^2 \)[/tex].
40. Divide [tex]\( \left(-x^6 + x^5 + 7x^2 - 9\right) \div x^4 \)[/tex]:
- Divide each term by [tex]\( x^4 \)[/tex]:
[tex]\[ \frac{-x^6}{x^4} = -x^2 \][/tex]
[tex]\[ \frac{x^5}{x^4} = x \][/tex]
[tex]\[ \frac{7x^2}{x^4} = \frac{7}{x^2} \text{ which remains in the solution} \][/tex]
[tex]\[ \frac{-9}{x^4} = -\frac{9}{x^4} \][/tex]
- The result is [tex]\( -x^2 + x + \frac{7x^2 - 9}{x^4} \)[/tex].
41. Divide [tex]\( \left(x^2 + 2x + 6\right) \div x \)[/tex]:
- Divide each term by [tex]\( x \)[/tex]:
[tex]\[ \frac{x^2}{x} = x \][/tex]
[tex]\[ \frac{2x}{x} = 2 \][/tex]
[tex]\[ \frac{6}{x} \text{ remains } 6 \][/tex]
- The result is [tex]\( x + 2 + \frac{6}{x} \)[/tex].
42. Divide [tex]\( \left(3x^2 - 15x + 5\right) \div (-3x) \)[/tex]:
- Divide each term by [tex]\( -3x \)[/tex]:
[tex]\[ \frac{3x^2}{-3x} = -x \][/tex]
[tex]\[ \frac{-15x}{-3x} = 5 \][/tex]
[tex]\[ \frac{5}{-3x} \text{ remains as a remainder} \][/tex]
- The result is [tex]\( -x + 5 + \frac{5}{-3x} \)[/tex].
43. Divide [tex]\( \left(2x^{11} - 5x^7 - 10x^6\right) \div 2x^3 \)[/tex]:
- Divide each term by [tex]\( 2x^3 \)[/tex]:
[tex]\[ \frac{2x^{11}}{2x^3} = x^8 \][/tex]
[tex]\[ \frac{-5x^7}{2x^3} = -\frac{5}{2}x^4 \][/tex]
[tex]\[ \frac{-10x^6}{2x^3} = -5x^3 \][/tex]
- The result is [tex]\( x^8 - \frac{5}{2}x^4 - 5x^3 \)[/tex].
44. Divide [tex]\( \left(-2x^6 + 5x^5 + 9x^2 + 2\right) \div x^4 \)[/tex]:
- Divide each term by [tex]\( x^4 \)[/tex]:
[tex]\[ \frac{-2x^6}{x^4} = -2x^2 \][/tex]
[tex]\[ \frac{5x^5}{x^4} = 5x \][/tex]
[tex]\[ \frac{9x^2}{x^4} = \frac{9}{x^2} \][/tex]
[tex]\[ \frac{2}{x^4} = \frac{2}{x^4} \][/tex]
- The result is [tex]\( -2x^2 + 5x + \frac{9x^2 + 2}{x^4} \)[/tex].
These steps help understand how each polynomial division is carried out. If you need further clarification on any specific step, feel free to ask!
