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Sagot :
Certainly! Let's go through each part step-by-step.
### Part (i)
Simplify: [tex]\(\left(8 x^3-4 x^2+5\right) \times \frac{1}{2} x+\left(3 x^2-5 x+1\right) \times 4 x\)[/tex]
1. Distribute [tex]\(\frac{1}{2} x\)[/tex] in the first term:
[tex]\[ (8 x^3 - 4 x^2 + 5) \times \frac{1}{2} x = 8 x^3 \times \frac{1}{2} x - 4 x^2 \times \frac{1}{2} x + 5 \times \frac{1}{2} x \][/tex]
[tex]\[ = 4 x^4 - 2 x^3 + \frac{5}{2} x \][/tex]
2. Distribute [tex]\(4 x\)[/tex] in the second term:
[tex]\[ (3 x^2 - 5 x + 1) \times 4 x = 3 x^2 \times 4 x - 5 x \times 4 x + 1 \times 4 x \][/tex]
[tex]\[ = 12 x^3 - 20 x^2 + 4 x \][/tex]
3. Combine both results:
[tex]\[ 4 x^4 - 2 x^3 + \frac{5}{2} x + 12 x^3 - 20 x^2 + 4 x \][/tex]
4. Simplify by combining like terms:
[tex]\[ = 4 x^4 + (12 x^3 - 2 x^3) - 20 x^2 + \left( \frac{5}{2} x + 4 x \right) \][/tex]
[tex]\[ = 4 x^4 + 10 x^3 - 20 x^2 + \frac{13}{2} x \][/tex]
### Part (ii)
Simplify: [tex]\(\left(5 a^3-18 a^2-25 a\right) \times \frac{1}{5} a-\left(9 a^2-6 a+11\right) \times \frac{-2}{3} a\)[/tex]
1. Distribute [tex]\(\frac{1}{5} a\)[/tex] in the first term:
[tex]\[ (5 a^3 - 18 a^2 - 25 a) \times \frac{1}{5} a = 5 a^3 \times \frac{1}{5} a - 18 a^2 \times \frac{1}{5} a - 25 a \times \frac{1}{5} a \][/tex]
[tex]\[ = a^4 - \frac{18}{5} a^3 - 5 a^2 \][/tex]
2. Distribute [tex]\(\frac{-2}{3} a\)[/tex] in the second term:
[tex]\[ (9 a^2 - 6 a + 11) \times \frac{-2}{3} a = 9 a^2 \times \frac{-2}{3} a - 6 a \times \frac{-2}{3} a + 11 \times \frac{-2}{3} a \][/tex]
[tex]\[ = -6 a^3 + 4 a^2 - \frac{22}{3} a \][/tex]
3. Combine both results subtracting the second result from the first:
[tex]\[ a^4 - \frac{18}{5} a^3 - 5 a^2 - (-6 a^3 + 4 a^2 - \frac{22}{3} a) \][/tex]
4. Distribute the negative sign:
[tex]\[ a^4 - \frac{18}{5} a^3 - 5 a^2 + 6 a^3 - 4 a^2 + \frac{22}{3} a \][/tex]
5. Simplify by combining like terms:
[tex]\[ = a^4 + \left(6 a^3 - \frac{18}{5} a^3\right) - \left(5 a^2 + 4 a^2\right) + \frac{22}{3} a \][/tex]
We simplify the coefficients as:
[tex]\[ a^4 + \left(6 - \frac{18}{5}\right) a^3 - 9 a^2 + \frac{22}{3} a \][/tex]
[tex]\[ = a^4 + \left(\frac{30}{5} - \frac{18}{5}\right) a^3 - 9 a^2 + \frac{22}{3} a \][/tex]
[tex]\[ = a^4 + \frac{12}{5} a^3 - 9 a^2 + \frac{22}{3} a \][/tex]
### Part (iii)
Simplify: [tex]\(\left(x^2+x y+\frac{1}{2} x y^2\right) \times 2 y+\left(y^2+2 x y-x^2\right) \times \frac{3}{2} x\)[/tex]
1. Distribute [tex]\(2 y\)[/tex] in the first term:
[tex]\[ \left( x^2 + x y + \frac{1}{2} x y^2 \right) \times 2 y = x^2 \times 2 y + x y \times 2 y + \frac{1}{2} x y^2 \times 2 y \][/tex]
[tex]\[ = 2 x^2 y + 2 x y^2 + x y^3 \][/tex]
2. Distribute [tex]\(\frac{3}{2} x\)[/tex] in the second term:
[tex]\[ \left( y^2 + 2 x y - x^2 \right) \times \frac{3}{2} x = y^2 \times \frac{3}{2} x + 2 x y \times \frac{3}{2} x - x^2 \times \frac{3}{2} x \][/tex]
[tex]\[ = \frac{3}{2} x y^2 + 3 x^2 y - \frac{3}{2} x^3 \][/tex]
3. Combine both results:
[tex]\[ 2 x^2 y + 2 x y^2 + x y^3 + \frac{3}{2} x y^2 + 3 x^2 y - \frac{3}{2} x^3 \][/tex]
4. Simplify by combining like terms:
[tex]\[ = (2 x^2 y + 3 x^2 y) + (2 x y^2 + \frac{3}{2} x y^2) + x y^3 - \frac{3}{2} x^3 \][/tex]
[tex]\[ = 5 x^2 y + \left(2 + \frac{3}{2}\right) x y^2 + x y^3 - \frac{3}{2} x^3 \][/tex]
[tex]\[ = 5 x^2 y + \frac{7}{2} x y^2 + x y^3 - \frac{3}{2} x^3 \][/tex]
### Part (iv)
Simplify: [tex]\(\left(9 x^2-12 x y+4 y^2\right) \times x y^2-\left(4 x^2-12 y^2+9 x y\right) \times x^2 y\)[/tex]
1. Distribute [tex]\(x y^2\)[/tex] in the first term:
[tex]\[ \left(9 x^2 - 12 x y + 4 y^2\right) \times x y^2 \][/tex]
[tex]\[ = 9 x^2 \times x y^2 - 12 x y \times x y^2 + 4 y^2 \times x y^2 \][/tex]
[tex]\[ = 9 x^3 y^2 - 12 x^2 y^3 + 4 x y^4 \][/tex]
2. Distribute [tex]\(x^2 y\)[/tex] in the second term:
[tex]\[ \left(4 x^2 - 12 y^2 + 9 x y\right) \times x^2 y \][/tex]
[tex]\[ = 4 x^2 \times x^2 y - 12 y^2 \times x^2 y + 9 x y \times x^2 y \][/tex]
[tex]\[ = 4 x^4 y - 12 x^2 y^3 + 9 x^3 y^2 \][/tex]
3. Combine both results subtracting the second from the first:
[tex]\[ 9 x^3 y^2 - 12 x^2 y^3 + 4 x y^4 - (4 x^4 y - 12 x^2 y^3 + 9 x^3 y^2) \][/tex]
4. Distribute the negative sign:
[tex]\[ 9 x^3 y^2 - 12 x^2 y^3 + 4 x y^4 - 4 x^4 y + 12 x^2 y^3 - 9 x^3 y^2 \][/tex]
5. Simplify by combining like terms:
[tex]\[ = (9 x^3 y^2 - 9 x^3 y^2) + (-12 x^2 y^3 + 12 x^2 y^3) + 4 x y^4 - 4 x^4 y \][/tex]
[tex]\[ = 4 x y^4 - 4 x^4 y \][/tex]
Thus, the simplified form is:
[tex]\[ 4 x y^4 - 4 x^4 y \][/tex]
So, the simplified answers are:
(i) [tex]\(4 x^4 + 10 x^3 - 20 x^2 + \frac{13}{2} x\)[/tex]
(ii) [tex]\(a^4 + \frac{12}{5} a^3 - 9 a^2 + \frac{22}{3} a\)[/tex]
(iii) [tex]\(5 x^2 y + \frac{7}{2} x y^2 + x y^3 - \frac{3}{2} x^3\)[/tex]
(iv) [tex]\(4 x y^4 - 4 x^4 y\)[/tex]
### Part (i)
Simplify: [tex]\(\left(8 x^3-4 x^2+5\right) \times \frac{1}{2} x+\left(3 x^2-5 x+1\right) \times 4 x\)[/tex]
1. Distribute [tex]\(\frac{1}{2} x\)[/tex] in the first term:
[tex]\[ (8 x^3 - 4 x^2 + 5) \times \frac{1}{2} x = 8 x^3 \times \frac{1}{2} x - 4 x^2 \times \frac{1}{2} x + 5 \times \frac{1}{2} x \][/tex]
[tex]\[ = 4 x^4 - 2 x^3 + \frac{5}{2} x \][/tex]
2. Distribute [tex]\(4 x\)[/tex] in the second term:
[tex]\[ (3 x^2 - 5 x + 1) \times 4 x = 3 x^2 \times 4 x - 5 x \times 4 x + 1 \times 4 x \][/tex]
[tex]\[ = 12 x^3 - 20 x^2 + 4 x \][/tex]
3. Combine both results:
[tex]\[ 4 x^4 - 2 x^3 + \frac{5}{2} x + 12 x^3 - 20 x^2 + 4 x \][/tex]
4. Simplify by combining like terms:
[tex]\[ = 4 x^4 + (12 x^3 - 2 x^3) - 20 x^2 + \left( \frac{5}{2} x + 4 x \right) \][/tex]
[tex]\[ = 4 x^4 + 10 x^3 - 20 x^2 + \frac{13}{2} x \][/tex]
### Part (ii)
Simplify: [tex]\(\left(5 a^3-18 a^2-25 a\right) \times \frac{1}{5} a-\left(9 a^2-6 a+11\right) \times \frac{-2}{3} a\)[/tex]
1. Distribute [tex]\(\frac{1}{5} a\)[/tex] in the first term:
[tex]\[ (5 a^3 - 18 a^2 - 25 a) \times \frac{1}{5} a = 5 a^3 \times \frac{1}{5} a - 18 a^2 \times \frac{1}{5} a - 25 a \times \frac{1}{5} a \][/tex]
[tex]\[ = a^4 - \frac{18}{5} a^3 - 5 a^2 \][/tex]
2. Distribute [tex]\(\frac{-2}{3} a\)[/tex] in the second term:
[tex]\[ (9 a^2 - 6 a + 11) \times \frac{-2}{3} a = 9 a^2 \times \frac{-2}{3} a - 6 a \times \frac{-2}{3} a + 11 \times \frac{-2}{3} a \][/tex]
[tex]\[ = -6 a^3 + 4 a^2 - \frac{22}{3} a \][/tex]
3. Combine both results subtracting the second result from the first:
[tex]\[ a^4 - \frac{18}{5} a^3 - 5 a^2 - (-6 a^3 + 4 a^2 - \frac{22}{3} a) \][/tex]
4. Distribute the negative sign:
[tex]\[ a^4 - \frac{18}{5} a^3 - 5 a^2 + 6 a^3 - 4 a^2 + \frac{22}{3} a \][/tex]
5. Simplify by combining like terms:
[tex]\[ = a^4 + \left(6 a^3 - \frac{18}{5} a^3\right) - \left(5 a^2 + 4 a^2\right) + \frac{22}{3} a \][/tex]
We simplify the coefficients as:
[tex]\[ a^4 + \left(6 - \frac{18}{5}\right) a^3 - 9 a^2 + \frac{22}{3} a \][/tex]
[tex]\[ = a^4 + \left(\frac{30}{5} - \frac{18}{5}\right) a^3 - 9 a^2 + \frac{22}{3} a \][/tex]
[tex]\[ = a^4 + \frac{12}{5} a^3 - 9 a^2 + \frac{22}{3} a \][/tex]
### Part (iii)
Simplify: [tex]\(\left(x^2+x y+\frac{1}{2} x y^2\right) \times 2 y+\left(y^2+2 x y-x^2\right) \times \frac{3}{2} x\)[/tex]
1. Distribute [tex]\(2 y\)[/tex] in the first term:
[tex]\[ \left( x^2 + x y + \frac{1}{2} x y^2 \right) \times 2 y = x^2 \times 2 y + x y \times 2 y + \frac{1}{2} x y^2 \times 2 y \][/tex]
[tex]\[ = 2 x^2 y + 2 x y^2 + x y^3 \][/tex]
2. Distribute [tex]\(\frac{3}{2} x\)[/tex] in the second term:
[tex]\[ \left( y^2 + 2 x y - x^2 \right) \times \frac{3}{2} x = y^2 \times \frac{3}{2} x + 2 x y \times \frac{3}{2} x - x^2 \times \frac{3}{2} x \][/tex]
[tex]\[ = \frac{3}{2} x y^2 + 3 x^2 y - \frac{3}{2} x^3 \][/tex]
3. Combine both results:
[tex]\[ 2 x^2 y + 2 x y^2 + x y^3 + \frac{3}{2} x y^2 + 3 x^2 y - \frac{3}{2} x^3 \][/tex]
4. Simplify by combining like terms:
[tex]\[ = (2 x^2 y + 3 x^2 y) + (2 x y^2 + \frac{3}{2} x y^2) + x y^3 - \frac{3}{2} x^3 \][/tex]
[tex]\[ = 5 x^2 y + \left(2 + \frac{3}{2}\right) x y^2 + x y^3 - \frac{3}{2} x^3 \][/tex]
[tex]\[ = 5 x^2 y + \frac{7}{2} x y^2 + x y^3 - \frac{3}{2} x^3 \][/tex]
### Part (iv)
Simplify: [tex]\(\left(9 x^2-12 x y+4 y^2\right) \times x y^2-\left(4 x^2-12 y^2+9 x y\right) \times x^2 y\)[/tex]
1. Distribute [tex]\(x y^2\)[/tex] in the first term:
[tex]\[ \left(9 x^2 - 12 x y + 4 y^2\right) \times x y^2 \][/tex]
[tex]\[ = 9 x^2 \times x y^2 - 12 x y \times x y^2 + 4 y^2 \times x y^2 \][/tex]
[tex]\[ = 9 x^3 y^2 - 12 x^2 y^3 + 4 x y^4 \][/tex]
2. Distribute [tex]\(x^2 y\)[/tex] in the second term:
[tex]\[ \left(4 x^2 - 12 y^2 + 9 x y\right) \times x^2 y \][/tex]
[tex]\[ = 4 x^2 \times x^2 y - 12 y^2 \times x^2 y + 9 x y \times x^2 y \][/tex]
[tex]\[ = 4 x^4 y - 12 x^2 y^3 + 9 x^3 y^2 \][/tex]
3. Combine both results subtracting the second from the first:
[tex]\[ 9 x^3 y^2 - 12 x^2 y^3 + 4 x y^4 - (4 x^4 y - 12 x^2 y^3 + 9 x^3 y^2) \][/tex]
4. Distribute the negative sign:
[tex]\[ 9 x^3 y^2 - 12 x^2 y^3 + 4 x y^4 - 4 x^4 y + 12 x^2 y^3 - 9 x^3 y^2 \][/tex]
5. Simplify by combining like terms:
[tex]\[ = (9 x^3 y^2 - 9 x^3 y^2) + (-12 x^2 y^3 + 12 x^2 y^3) + 4 x y^4 - 4 x^4 y \][/tex]
[tex]\[ = 4 x y^4 - 4 x^4 y \][/tex]
Thus, the simplified form is:
[tex]\[ 4 x y^4 - 4 x^4 y \][/tex]
So, the simplified answers are:
(i) [tex]\(4 x^4 + 10 x^3 - 20 x^2 + \frac{13}{2} x\)[/tex]
(ii) [tex]\(a^4 + \frac{12}{5} a^3 - 9 a^2 + \frac{22}{3} a\)[/tex]
(iii) [tex]\(5 x^2 y + \frac{7}{2} x y^2 + x y^3 - \frac{3}{2} x^3\)[/tex]
(iv) [tex]\(4 x y^4 - 4 x^4 y\)[/tex]
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