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Sagot :
Sure, let's solve each of these polynomial divisions step-by-step:
### (i) Divide [tex]\( x^2 - 9x + 20 \)[/tex] by [tex]\( x - 5 \)[/tex]:
To perform the division, we can use polynomial long division or simply recognize that [tex]\( x^2 - 9x + 20 \)[/tex] can be factored.
[tex]\[ x^2 - 9x + 20 = (x - 5)(x - 4) \][/tex]
Therefore, dividing [tex]\( x^2 - 9x + 20 \)[/tex] by [tex]\( x - 5 \)[/tex]:
[tex]\[ \frac{x^2 - 9x + 20}{x - 5} = x - 4 \][/tex]
### (ii) Divide [tex]\( x^2 - 11x - 42 \)[/tex] by [tex]\( x + 3 \)[/tex]:
[tex]\[ x^2 - 11x - 42 \][/tex] can be factored as:
[tex]\[ x^2 - 11x - 42 = (x + 3)(x - 14) \][/tex]
Thus, dividing by [tex]\( x + 3 \)[/tex]:
[tex]\[ \frac{x^2 - 11x - 42}{x + 3} = x - 14 \][/tex]
### (iii) Divide [tex]\( a^2 - 6a + 9 \)[/tex] by [tex]\( a - 3 \)[/tex]:
The polynomial [tex]\( a^2 - 6a + 9 \)[/tex] is a perfect square:
[tex]\[ a^2 - 6a + 9 = (a - 3)^2 \][/tex]
So dividing [tex]\( (a - 3)^2 \)[/tex] by [tex]\( a - 3 \)[/tex]:
[tex]\[ \frac{a^2 - 6a + 9}{a - 3} = a - 3 \][/tex]
### (iv) Divide [tex]\( y^2 + 21y + 54 \)[/tex] by [tex]\( y + 3 \)[/tex]:
The polynomial [tex]\( y^2 + 21y + 54 \)[/tex] factors as:
[tex]\[ y^2 + 21y + 54 = (y + 3)(y + 18) \][/tex]
So dividing by [tex]\( y + 3 \)[/tex]:
[tex]\[ \frac{y^2 + 21y + 54}{y + 3} = y + 18 \][/tex]
### (v) Divide [tex]\( x^2 - 25x + 66 \)[/tex] by [tex]\( x - 3 \)[/tex]:
Factoring [tex]\( x^2 - 25x + 66 \)[/tex]:
[tex]\[ x^2 - 25x + 66 = (x - 3)(x - 22) \][/tex]
Thus dividing by [tex]\( x - 3 \)[/tex]:
[tex]\[ \frac{x^2 - 25x + 66}{x - 3} = x - 22 \][/tex]
### (vi) Divide [tex]\( x^2 - 12x + 35 \)[/tex] by [tex]\( x - 7 \)[/tex]:
Factoring [tex]\( x^2 - 12x + 35 \)[/tex]:
[tex]\[ x^2 - 12x + 35 = (x - 7)(x - 5) \][/tex]
So dividing by [tex]\( x - 7 \)[/tex]:
[tex]\[ \frac{x^2 - 12x + 35}{x - 7} = x - 5 \][/tex]
### (vii) Divide [tex]\( 14x^2 + 19x - 3 \)[/tex] by [tex]\( 7x - 1 \)[/tex]:
This division seems a bit more involved, but we can find the quotient:
[tex]\[ 14x^2 + 19x - 3 = (7x - 1)(2x + 3) \][/tex]
Thus:
[tex]\[ \frac{14x^2 + 19x - 3}{7x - 1} = 2x + 3 \][/tex]
### (viii) Divide [tex]\( x^2 - 7xy + 12y^2 \)[/tex] by [tex]\( x - 3y \)[/tex]:
Factoring [tex]\( x^2 - 7xy + 12y^2 \)[/tex]:
[tex]\[ x^2 - 7xy + 12y^2 = (x - 3y)(x - 4y) \][/tex]
So dividing by [tex]\( x - 3y \)[/tex]:
[tex]\[ \frac{x^2 - 7xy + 12y^2}{x - 3y} = x - 4y \][/tex]
### Summary of Results:
So, the final results for each division problem are:
1. [tex]\( \frac{x^2 - 9x + 20}{x - 5} = x - 4 \)[/tex]
2. [tex]\( \frac{x^2 - 11x - 42}{x + 3} = x - 14 \)[/tex]
3. [tex]\( \frac{a^2 - 6a + 9}{a - 3} = a - 3 \)[/tex]
4. [tex]\( \frac{y^2 + 21y + 54}{y + 3} = y + 18 \)[/tex]
5. [tex]\( \frac{x^2 - 25x + 66}{x - 3} = x - 22 \)[/tex]
6. [tex]\( \frac{x^2 - 12x + 35}{x - 7} = x - 5 \)[/tex]
7. [tex]\( \frac{14x^2 + 19x - 3}{7x - 1} = 2x + 3 \)[/tex]
8. [tex]\( \frac{x^2 - 7xy + 12y^2}{x - 3y} = x - 4y \)[/tex]
### (i) Divide [tex]\( x^2 - 9x + 20 \)[/tex] by [tex]\( x - 5 \)[/tex]:
To perform the division, we can use polynomial long division or simply recognize that [tex]\( x^2 - 9x + 20 \)[/tex] can be factored.
[tex]\[ x^2 - 9x + 20 = (x - 5)(x - 4) \][/tex]
Therefore, dividing [tex]\( x^2 - 9x + 20 \)[/tex] by [tex]\( x - 5 \)[/tex]:
[tex]\[ \frac{x^2 - 9x + 20}{x - 5} = x - 4 \][/tex]
### (ii) Divide [tex]\( x^2 - 11x - 42 \)[/tex] by [tex]\( x + 3 \)[/tex]:
[tex]\[ x^2 - 11x - 42 \][/tex] can be factored as:
[tex]\[ x^2 - 11x - 42 = (x + 3)(x - 14) \][/tex]
Thus, dividing by [tex]\( x + 3 \)[/tex]:
[tex]\[ \frac{x^2 - 11x - 42}{x + 3} = x - 14 \][/tex]
### (iii) Divide [tex]\( a^2 - 6a + 9 \)[/tex] by [tex]\( a - 3 \)[/tex]:
The polynomial [tex]\( a^2 - 6a + 9 \)[/tex] is a perfect square:
[tex]\[ a^2 - 6a + 9 = (a - 3)^2 \][/tex]
So dividing [tex]\( (a - 3)^2 \)[/tex] by [tex]\( a - 3 \)[/tex]:
[tex]\[ \frac{a^2 - 6a + 9}{a - 3} = a - 3 \][/tex]
### (iv) Divide [tex]\( y^2 + 21y + 54 \)[/tex] by [tex]\( y + 3 \)[/tex]:
The polynomial [tex]\( y^2 + 21y + 54 \)[/tex] factors as:
[tex]\[ y^2 + 21y + 54 = (y + 3)(y + 18) \][/tex]
So dividing by [tex]\( y + 3 \)[/tex]:
[tex]\[ \frac{y^2 + 21y + 54}{y + 3} = y + 18 \][/tex]
### (v) Divide [tex]\( x^2 - 25x + 66 \)[/tex] by [tex]\( x - 3 \)[/tex]:
Factoring [tex]\( x^2 - 25x + 66 \)[/tex]:
[tex]\[ x^2 - 25x + 66 = (x - 3)(x - 22) \][/tex]
Thus dividing by [tex]\( x - 3 \)[/tex]:
[tex]\[ \frac{x^2 - 25x + 66}{x - 3} = x - 22 \][/tex]
### (vi) Divide [tex]\( x^2 - 12x + 35 \)[/tex] by [tex]\( x - 7 \)[/tex]:
Factoring [tex]\( x^2 - 12x + 35 \)[/tex]:
[tex]\[ x^2 - 12x + 35 = (x - 7)(x - 5) \][/tex]
So dividing by [tex]\( x - 7 \)[/tex]:
[tex]\[ \frac{x^2 - 12x + 35}{x - 7} = x - 5 \][/tex]
### (vii) Divide [tex]\( 14x^2 + 19x - 3 \)[/tex] by [tex]\( 7x - 1 \)[/tex]:
This division seems a bit more involved, but we can find the quotient:
[tex]\[ 14x^2 + 19x - 3 = (7x - 1)(2x + 3) \][/tex]
Thus:
[tex]\[ \frac{14x^2 + 19x - 3}{7x - 1} = 2x + 3 \][/tex]
### (viii) Divide [tex]\( x^2 - 7xy + 12y^2 \)[/tex] by [tex]\( x - 3y \)[/tex]:
Factoring [tex]\( x^2 - 7xy + 12y^2 \)[/tex]:
[tex]\[ x^2 - 7xy + 12y^2 = (x - 3y)(x - 4y) \][/tex]
So dividing by [tex]\( x - 3y \)[/tex]:
[tex]\[ \frac{x^2 - 7xy + 12y^2}{x - 3y} = x - 4y \][/tex]
### Summary of Results:
So, the final results for each division problem are:
1. [tex]\( \frac{x^2 - 9x + 20}{x - 5} = x - 4 \)[/tex]
2. [tex]\( \frac{x^2 - 11x - 42}{x + 3} = x - 14 \)[/tex]
3. [tex]\( \frac{a^2 - 6a + 9}{a - 3} = a - 3 \)[/tex]
4. [tex]\( \frac{y^2 + 21y + 54}{y + 3} = y + 18 \)[/tex]
5. [tex]\( \frac{x^2 - 25x + 66}{x - 3} = x - 22 \)[/tex]
6. [tex]\( \frac{x^2 - 12x + 35}{x - 7} = x - 5 \)[/tex]
7. [tex]\( \frac{14x^2 + 19x - 3}{7x - 1} = 2x + 3 \)[/tex]
8. [tex]\( \frac{x^2 - 7xy + 12y^2}{x - 3y} = x - 4y \)[/tex]
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