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Sagot :
Let's go through the divisions step-by-step:
### (a) [tex]\((m^2 - 14m - 32) \div (m + 2)\)[/tex]
1. Set up the division:
[tex]\[ \frac{m^2 - 14m - 32}{m + 2} \][/tex]
2. Perform synthetic division or polynomial long division to find the quotient.
The quotient is:
[tex]\[ m - 16 \][/tex]
So, [tex]\[ \left(m^2-14 m-32\right) \div(m+2) = m - 16 \][/tex]
### (b) [tex]\(4yz(z^2 + 6z - 16) \div 2y(z + 8)\)[/tex]
1. Simplify the expression:
Simplify the constants and factors:
[tex]\[ \frac{4yz(z^2 + 6z - 16)}{2y(z + 8)} = 2z\frac{z^2 + 6z - 16}{z + 8} \][/tex]
2. Divide the polynomial:
[tex]\[ \frac{z^2 + 6z - 16}{z + 8} = z - 2 \][/tex]
So, [tex]\[ 4 y z\left(z^2+6 z-16\right) \div 2 y(z+8) = 2z(z - 2) = 2z^2 - 4z \][/tex]
### (c) [tex]\(39y^3(50y^2 - 98) \div 26y^2(5y + 7)\)[/tex]
1. Simplify the expression:
Simplify the coefficients and common factors:
[tex]\[ \frac{39y^3(50y^2 - 98)}{26y^2(5y + 7)} = \frac{3}{2} \cdot y \cdot y^2(50y^2 - 98) \div (5y + 7) \][/tex]
2. Divide the polynomial part:
[tex]\[ \frac{50y^2 - 98}{5y + 7} = 10y - 14 \][/tex]
So,
[tex]\[ 39 y^3\left(50 y^2-98\right) \div 26 y^2(5 y+7) = 15y^2 - 21y \][/tex]
### (d) [tex]\(24ab(9a^2 - 16b^2) \div 8ab(3a + 4b)\)[/tex]
1. Simplify the expression:
[tex]\[ \frac{24ab(9a^2 - 16b^2)}{8ab(3a + 4b)} = 3 \cdot a \cdot b \cdot \frac{9a^2 - 16b^2}{3a + 4b} \][/tex]
2. Divide the polynomial part:
[tex]\[ \frac{9a^2 - 16b^2}{3a + 4b} = 3a - 4b \][/tex]
So,
[tex]\[ 24 ab\left(9 a^2-16 b^2\right) \div 8 ab(3 a+4 b) = 9a - 12b \][/tex]
### (e) [tex]\((x^2 + 7x + 10) \div (x + 5)\)[/tex]
1. Set up the division:
[tex]\[ \frac{x^2 + 7x + 10}{x + 5} \][/tex]
2. Perform synthetic division or polynomial long division:
The quotient is:
[tex]\[ x + 2 \][/tex]
So,
[tex]\[ \left(x^2+7 x+10\right) \div (x+5) = x + 2 \][/tex]
### (f) [tex]\((y^2 - y - 12) \div (y - 4)\)[/tex]
1. Set up the division:
[tex]\[ \frac{y^2 - y - 12}{y - 4} \][/tex]
2. Present the quotient:
[tex]\[ y + 3 \][/tex]
So,
[tex]\[ \left(y^2 - y - 12\right) \div (y - 4) = y + 3 \][/tex]
### (g) [tex]\((y^2 - 5y + 6) \div (y - 2)\)[/tex]
1. Set up the division:
[tex]\[ \frac{y^2 - 5y + 6}{y - 2} \][/tex]
2. Present the quotient:
[tex]\[ y - 3 \][/tex]
So,
[tex]\[ \left(y^2-5 y+6\right) \div (y-2) = y - 3 \][/tex]
### (h) [tex]\((6x^2 - 5x - 6) \div (3x + 2)\)[/tex]
1. Set up the division:
[tex]\[ \frac{6x^2 - 5x - 6}{3x + 2} \][/tex]
2. Present the quotient:
[tex]\[ 2x - 3 \][/tex]
So,
[tex]\[ \left(6 x^2-5 x-6\right) \div (3 x+2) = 2x - 3 \][/tex]
These steps give the required results for each division.
### (a) [tex]\((m^2 - 14m - 32) \div (m + 2)\)[/tex]
1. Set up the division:
[tex]\[ \frac{m^2 - 14m - 32}{m + 2} \][/tex]
2. Perform synthetic division or polynomial long division to find the quotient.
The quotient is:
[tex]\[ m - 16 \][/tex]
So, [tex]\[ \left(m^2-14 m-32\right) \div(m+2) = m - 16 \][/tex]
### (b) [tex]\(4yz(z^2 + 6z - 16) \div 2y(z + 8)\)[/tex]
1. Simplify the expression:
Simplify the constants and factors:
[tex]\[ \frac{4yz(z^2 + 6z - 16)}{2y(z + 8)} = 2z\frac{z^2 + 6z - 16}{z + 8} \][/tex]
2. Divide the polynomial:
[tex]\[ \frac{z^2 + 6z - 16}{z + 8} = z - 2 \][/tex]
So, [tex]\[ 4 y z\left(z^2+6 z-16\right) \div 2 y(z+8) = 2z(z - 2) = 2z^2 - 4z \][/tex]
### (c) [tex]\(39y^3(50y^2 - 98) \div 26y^2(5y + 7)\)[/tex]
1. Simplify the expression:
Simplify the coefficients and common factors:
[tex]\[ \frac{39y^3(50y^2 - 98)}{26y^2(5y + 7)} = \frac{3}{2} \cdot y \cdot y^2(50y^2 - 98) \div (5y + 7) \][/tex]
2. Divide the polynomial part:
[tex]\[ \frac{50y^2 - 98}{5y + 7} = 10y - 14 \][/tex]
So,
[tex]\[ 39 y^3\left(50 y^2-98\right) \div 26 y^2(5 y+7) = 15y^2 - 21y \][/tex]
### (d) [tex]\(24ab(9a^2 - 16b^2) \div 8ab(3a + 4b)\)[/tex]
1. Simplify the expression:
[tex]\[ \frac{24ab(9a^2 - 16b^2)}{8ab(3a + 4b)} = 3 \cdot a \cdot b \cdot \frac{9a^2 - 16b^2}{3a + 4b} \][/tex]
2. Divide the polynomial part:
[tex]\[ \frac{9a^2 - 16b^2}{3a + 4b} = 3a - 4b \][/tex]
So,
[tex]\[ 24 ab\left(9 a^2-16 b^2\right) \div 8 ab(3 a+4 b) = 9a - 12b \][/tex]
### (e) [tex]\((x^2 + 7x + 10) \div (x + 5)\)[/tex]
1. Set up the division:
[tex]\[ \frac{x^2 + 7x + 10}{x + 5} \][/tex]
2. Perform synthetic division or polynomial long division:
The quotient is:
[tex]\[ x + 2 \][/tex]
So,
[tex]\[ \left(x^2+7 x+10\right) \div (x+5) = x + 2 \][/tex]
### (f) [tex]\((y^2 - y - 12) \div (y - 4)\)[/tex]
1. Set up the division:
[tex]\[ \frac{y^2 - y - 12}{y - 4} \][/tex]
2. Present the quotient:
[tex]\[ y + 3 \][/tex]
So,
[tex]\[ \left(y^2 - y - 12\right) \div (y - 4) = y + 3 \][/tex]
### (g) [tex]\((y^2 - 5y + 6) \div (y - 2)\)[/tex]
1. Set up the division:
[tex]\[ \frac{y^2 - 5y + 6}{y - 2} \][/tex]
2. Present the quotient:
[tex]\[ y - 3 \][/tex]
So,
[tex]\[ \left(y^2-5 y+6\right) \div (y-2) = y - 3 \][/tex]
### (h) [tex]\((6x^2 - 5x - 6) \div (3x + 2)\)[/tex]
1. Set up the division:
[tex]\[ \frac{6x^2 - 5x - 6}{3x + 2} \][/tex]
2. Present the quotient:
[tex]\[ 2x - 3 \][/tex]
So,
[tex]\[ \left(6 x^2-5 x-6\right) \div (3 x+2) = 2x - 3 \][/tex]
These steps give the required results for each division.
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