Answer :
Certainly! Let's go through the problem step-by-step in detail for each expression.
### (viii) Expression:
[tex]\[ \left(2 x^3 + 7 x + 10 x^2\right) \times \frac{5}{2} x \][/tex]
[tex]\[ \begin{align*} &= (2 x^3 + 10 x^2 + 7 x) \times \frac{5}{2} x \\ &= 2 x^3 \times \frac{5}{2} x + 10 x^2 \times \frac{5}{2} x + 7 x \times \frac{5}{2} x \\ &= (2 \times \frac{5}{2} x^4) + (10 \times \frac{5}{2} x^3) + (7 \times \frac{5}{2} x^2) \\ &= 5 x^4 + 25 x^3 + 17.5 x^2 \\ &= x^2 \times (5 x^2 + 25 x + 17.5) \end{align*} \][/tex]
### (i) Expression:
[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]
[tex]\[ \begin{align*} &= \left(8 x^3 - 4 x^2 + 5\right) \times \frac{1}{2} x + \left(3 x^2 - 5 x + 1\right) \times 4 x \\ &= \left(8 x^3 \times \frac{1}{2} x - 4 x^2 \times \frac{1}{2} x + 5 \times \frac{1}{2} x\right) + \left(3 x^2 \times 4 x - 5 x \times 4 x + 1 \times 4 x\right) \\ &= \left(4 x^4 - 2 x^3 + 2.5 x\right) + \left(12 x^3 - 20 x^2 + 4 x\right) \\ &= 4 x^4 - 2 x^3 + 12 x^3 - 20 x^2 + 2.5 x + 4 x \\ &= 4 x^4 + 10 x^3 - 20 x^2 + 6.5 x \\ &= x \times (4 x^3 + 10 x^2 - 20 x + 6.5) \end{align*} \][/tex]
### (ii) Expression:
[tex]\[ \left(5 a^3 - 18 a^2 - 25 a\right) \times \frac{1}{5} a \times \left(9 a^2 - 6 a + 11\right) \times \frac{-2}{3} a \][/tex]
[tex]\[ \begin{align*} &= (5 a^3 - 18 a^2 - 25 a) \times \frac{1}{5} a \times (9 a^2 - 6 a + 11) \times \frac{-2}{3} a \\ &= \left(5 a^3 \times \frac{1}{5} a - 18 a^2 \times \frac{1}{5} a - 25 a \times \frac{1}{5} a\right) \times \left(9 a^2 \times \frac{-2}{3} a - 6 a \times \frac{-2}{3} a + 11 \times \frac{-2}{3} a\right)\\ &= (a^4 - 3.6 a^3 - 5 a^2) \times (-6 a^3 + 4 a^2 - 7.33333 a) \\ &= a^3 \times (-6 a^4 + 25.6 a^3 + 8.26667 a^2 + 6.4 a + 36.6667) \end{align*} \][/tex]
### (fii) Expression:
[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]
[tex]\[ \begin{align*} &= (x^2 + x y + \frac{1}{2} x y^2) \times 2 y + (y^2 + 2 x y - x^2) \times \frac{3}{2} x \\ &= \left(2 y x^2 + 2 x y^2 + x y \times y + \frac{1}{2} x (2 y^3)\right) + \left(\frac{3}{2} y^2 x + \frac{3}{2} \times 2 x y x - \frac{3}{2} x^3\right) \\ &= 2 x^2 y + 2 x y^2 + x y^2 + x y^3 + \frac{3}{2} x y^2 + 3 x^2 y - \frac{3}{2} x^3 \\ &= -1.5 x^3 + 5 x^2 y + 3.5 y^2 x + y^3 \end{align*} \][/tex]
### (iv) Expression:
[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]
[tex]\[ \begin{align*} &= \left(9 x^2 - 12 x y + 4 y^2\right) \times x y^2 - \left(4 x^2 - 12 y^2 + 9 x y\right) \times x^2 y \\ &= 9 x^3 y^2 - 12 x^2 y^3 + 4 y^4 x - 4 x^3 y^2 + 9 x^2 y^3 \\ &= -x^3 + y^3 \\ &= 4 x y \left(-x^3 + y^3\right) \end{align*} \][/tex]
I hope this step-by-step helps! If you have any further questions or need clarification, feel free to ask!
### (viii) Expression:
[tex]\[ \left(2 x^3 + 7 x + 10 x^2\right) \times \frac{5}{2} x \][/tex]
[tex]\[ \begin{align*} &= (2 x^3 + 10 x^2 + 7 x) \times \frac{5}{2} x \\ &= 2 x^3 \times \frac{5}{2} x + 10 x^2 \times \frac{5}{2} x + 7 x \times \frac{5}{2} x \\ &= (2 \times \frac{5}{2} x^4) + (10 \times \frac{5}{2} x^3) + (7 \times \frac{5}{2} x^2) \\ &= 5 x^4 + 25 x^3 + 17.5 x^2 \\ &= x^2 \times (5 x^2 + 25 x + 17.5) \end{align*} \][/tex]
### (i) Expression:
[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]
[tex]\[ \begin{align*} &= \left(8 x^3 - 4 x^2 + 5\right) \times \frac{1}{2} x + \left(3 x^2 - 5 x + 1\right) \times 4 x \\ &= \left(8 x^3 \times \frac{1}{2} x - 4 x^2 \times \frac{1}{2} x + 5 \times \frac{1}{2} x\right) + \left(3 x^2 \times 4 x - 5 x \times 4 x + 1 \times 4 x\right) \\ &= \left(4 x^4 - 2 x^3 + 2.5 x\right) + \left(12 x^3 - 20 x^2 + 4 x\right) \\ &= 4 x^4 - 2 x^3 + 12 x^3 - 20 x^2 + 2.5 x + 4 x \\ &= 4 x^4 + 10 x^3 - 20 x^2 + 6.5 x \\ &= x \times (4 x^3 + 10 x^2 - 20 x + 6.5) \end{align*} \][/tex]
### (ii) Expression:
[tex]\[ \left(5 a^3 - 18 a^2 - 25 a\right) \times \frac{1}{5} a \times \left(9 a^2 - 6 a + 11\right) \times \frac{-2}{3} a \][/tex]
[tex]\[ \begin{align*} &= (5 a^3 - 18 a^2 - 25 a) \times \frac{1}{5} a \times (9 a^2 - 6 a + 11) \times \frac{-2}{3} a \\ &= \left(5 a^3 \times \frac{1}{5} a - 18 a^2 \times \frac{1}{5} a - 25 a \times \frac{1}{5} a\right) \times \left(9 a^2 \times \frac{-2}{3} a - 6 a \times \frac{-2}{3} a + 11 \times \frac{-2}{3} a\right)\\ &= (a^4 - 3.6 a^3 - 5 a^2) \times (-6 a^3 + 4 a^2 - 7.33333 a) \\ &= a^3 \times (-6 a^4 + 25.6 a^3 + 8.26667 a^2 + 6.4 a + 36.6667) \end{align*} \][/tex]
### (fii) Expression:
[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]
[tex]\[ \begin{align*} &= (x^2 + x y + \frac{1}{2} x y^2) \times 2 y + (y^2 + 2 x y - x^2) \times \frac{3}{2} x \\ &= \left(2 y x^2 + 2 x y^2 + x y \times y + \frac{1}{2} x (2 y^3)\right) + \left(\frac{3}{2} y^2 x + \frac{3}{2} \times 2 x y x - \frac{3}{2} x^3\right) \\ &= 2 x^2 y + 2 x y^2 + x y^2 + x y^3 + \frac{3}{2} x y^2 + 3 x^2 y - \frac{3}{2} x^3 \\ &= -1.5 x^3 + 5 x^2 y + 3.5 y^2 x + y^3 \end{align*} \][/tex]
### (iv) Expression:
[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]
[tex]\[ \begin{align*} &= \left(9 x^2 - 12 x y + 4 y^2\right) \times x y^2 - \left(4 x^2 - 12 y^2 + 9 x y\right) \times x^2 y \\ &= 9 x^3 y^2 - 12 x^2 y^3 + 4 y^4 x - 4 x^3 y^2 + 9 x^2 y^3 \\ &= -x^3 + y^3 \\ &= 4 x y \left(-x^3 + y^3\right) \end{align*} \][/tex]
I hope this step-by-step helps! If you have any further questions or need clarification, feel free to ask!
