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Sagot :
Alright, let's simplify each of the given expressions step-by-step.
### Part (i)
Given:
[tex]\[ 2 q\left(3 p^2-3 p q + 8 \right) - 3 p(p - q) \][/tex]
Let's distribute and simplify the terms inside the parentheses:
1. Distribute [tex]\(2q\)[/tex]:
[tex]\[ 2q(3p^2) - 2q(3pq) + 2q(8) = 6pq^2 - 6pq^2 + 16q \][/tex]
2. Expand the other term:
[tex]\[ -3p(p - q) = -3p^2 + 3pq \][/tex]
Now combine the terms:
[tex]\[ 6qp^2 - 6pq^2 + 16q - 3p^2 + 3pq \][/tex]
Simplify by combining like terms:
[tex]\[ 2q(3p^2 - 3pq + 8) - 3p(p - q) = -3p(p - q) + 2q(3p^2 - 3pq + 8) \][/tex]
### Part (ii)
Given:
[tex]\[ m^2 n^2 (2 m - n^2 ) - m n^2 (4 m n - 2 m^2 ) + m^3 n (4 - 3 n) \][/tex]
Distribute within each term:
1. First term:
[tex]\[ m^2 n^2 (2m - n^2) = 2m^3 n^2 - m^2 n^4 \][/tex]
2. Second term:
[tex]\[ -mn^2 (4mn - 2m^2) = -4m^2 n^3 + 2m^3 n^2 \][/tex]
3. Third term:
[tex]\[ m^3 n (4 - 3n) = 4m^3 n - 3m^3 n^2 \][/tex]
Combine all terms:
[tex]\[ 2m^3 n^2 - m^2 n^4 - 4m^2 n^3 + 2m^3 n^2 + 4m^3 n - 3m^3 n^2 \][/tex]
Combine like terms:
[tex]\[ m^2 n^2 (2m - n^2) - mn^2 (4mn - 2m^2) + m^3 n (4 - 3n) = m^2n(mn + 4m - n^3 - 4n^2) \][/tex]
### Part (iii)
Given:
[tex]\[ (5a - 3b)(-a + 6b) - (2a + 3b)(3a - 4b) \][/tex]
Distribute each term:
1. First term:
[tex]\[ (5a - 3b)(-a + 6b) = 5a(-a + 6b) - 3b(-a + 6b) = -5a^2 + 30ab + 3ab - 18b^2 \][/tex]
2. Second term:
[tex]\[ (2a + 3b)(3a - 4b) = 2a(3a - 4b) + 3b(3a - 4b) = 6a^2 - 8ab + 9ab - 12b^2 \][/tex]
Combine the like terms:
[tex]\[ -11a^2 + 32ab - 6b^2 \][/tex]
### Part (iv)
Given:
[tex]\[ (x + y)(x - y - xy) + (x - y)(-x + y + xy) \][/tex]
Distribute each term:
1. First term:
[tex]\[ (x + y)(x - y - xy) = x(x - y - xy) + y(x - y - xy) = x^2 - xy - x^2y + yx - y^2 - xy^2 \][/tex]
2. Second term:
[tex]\[ (x - y)(-x + y + xy) = x(-x + y + xy) - y(-x + y + xy) = -x^2 + xy + x^2y - yx + y^2 - xy^2 \][/tex]
Combine the like terms:
[tex]\[ 2y(-xy + x - y) \][/tex]
So the simplified forms are:
(i) [tex]\(-3p(p - q) + 2q(3p^2 \- 3pq + 8)\)[/tex]
(ii) [tex]\(m^2n(mn + 4m - n^3 - 4n^2)\)[/tex]
(iii) [tex]\(-11a^2 + 32ab - 6b^2\)[/tex]
(iv) [tex]\(2y(-xy + x - y)\)[/tex]
### Part (i)
Given:
[tex]\[ 2 q\left(3 p^2-3 p q + 8 \right) - 3 p(p - q) \][/tex]
Let's distribute and simplify the terms inside the parentheses:
1. Distribute [tex]\(2q\)[/tex]:
[tex]\[ 2q(3p^2) - 2q(3pq) + 2q(8) = 6pq^2 - 6pq^2 + 16q \][/tex]
2. Expand the other term:
[tex]\[ -3p(p - q) = -3p^2 + 3pq \][/tex]
Now combine the terms:
[tex]\[ 6qp^2 - 6pq^2 + 16q - 3p^2 + 3pq \][/tex]
Simplify by combining like terms:
[tex]\[ 2q(3p^2 - 3pq + 8) - 3p(p - q) = -3p(p - q) + 2q(3p^2 - 3pq + 8) \][/tex]
### Part (ii)
Given:
[tex]\[ m^2 n^2 (2 m - n^2 ) - m n^2 (4 m n - 2 m^2 ) + m^3 n (4 - 3 n) \][/tex]
Distribute within each term:
1. First term:
[tex]\[ m^2 n^2 (2m - n^2) = 2m^3 n^2 - m^2 n^4 \][/tex]
2. Second term:
[tex]\[ -mn^2 (4mn - 2m^2) = -4m^2 n^3 + 2m^3 n^2 \][/tex]
3. Third term:
[tex]\[ m^3 n (4 - 3n) = 4m^3 n - 3m^3 n^2 \][/tex]
Combine all terms:
[tex]\[ 2m^3 n^2 - m^2 n^4 - 4m^2 n^3 + 2m^3 n^2 + 4m^3 n - 3m^3 n^2 \][/tex]
Combine like terms:
[tex]\[ m^2 n^2 (2m - n^2) - mn^2 (4mn - 2m^2) + m^3 n (4 - 3n) = m^2n(mn + 4m - n^3 - 4n^2) \][/tex]
### Part (iii)
Given:
[tex]\[ (5a - 3b)(-a + 6b) - (2a + 3b)(3a - 4b) \][/tex]
Distribute each term:
1. First term:
[tex]\[ (5a - 3b)(-a + 6b) = 5a(-a + 6b) - 3b(-a + 6b) = -5a^2 + 30ab + 3ab - 18b^2 \][/tex]
2. Second term:
[tex]\[ (2a + 3b)(3a - 4b) = 2a(3a - 4b) + 3b(3a - 4b) = 6a^2 - 8ab + 9ab - 12b^2 \][/tex]
Combine the like terms:
[tex]\[ -11a^2 + 32ab - 6b^2 \][/tex]
### Part (iv)
Given:
[tex]\[ (x + y)(x - y - xy) + (x - y)(-x + y + xy) \][/tex]
Distribute each term:
1. First term:
[tex]\[ (x + y)(x - y - xy) = x(x - y - xy) + y(x - y - xy) = x^2 - xy - x^2y + yx - y^2 - xy^2 \][/tex]
2. Second term:
[tex]\[ (x - y)(-x + y + xy) = x(-x + y + xy) - y(-x + y + xy) = -x^2 + xy + x^2y - yx + y^2 - xy^2 \][/tex]
Combine the like terms:
[tex]\[ 2y(-xy + x - y) \][/tex]
So the simplified forms are:
(i) [tex]\(-3p(p - q) + 2q(3p^2 \- 3pq + 8)\)[/tex]
(ii) [tex]\(m^2n(mn + 4m - n^3 - 4n^2)\)[/tex]
(iii) [tex]\(-11a^2 + 32ab - 6b^2\)[/tex]
(iv) [tex]\(2y(-xy + x - y)\)[/tex]
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